By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields
and their Galois closures
with 3 possible types to pure quintic fields
and their pure metacyclic normal fields
with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D<103
. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K)
of N over the cyclotomic field K=Q(ξ5)
, by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN))
by the number #(PN/K/PK)
of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K
. The precise structure of the
F5
-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (U<SUB>N</SUB>:U<SUB>0</SUB>).
The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.
Pure Quintic Fields Pure Metacyclic Fields Units Galois Cohomology Differential Principal Factorization Types Similarity Classes Prototypes Class Group Structure1. Introduction
At the end of his 1975 article on class numbers of pure quintic fields, Parry suggested verbatim: In conclusion, the author would like to say that he believes a numerical study of pure quintic fields would be most interesting ( [1] p. 484). Of course, it would have been rather difficult to realize Parry’s desire in 1975. But now, 40 years later, we are in the position to use the powerful computer algebra systems PARI/GP [2] and MAGMA [3] [4] [5] for starting an attack against this hard problem. Prepared by [6] [7] [8] [9] , this will actually be done in the present paper.
Even in 1991, when we generalized Barrucand and Cohn’s theory [10] [11] of principal factorization types from pure cubic fields to pure quintic fields and their pure metacyclic normal closures [12] , it was still impossible to verify our hypothesis about the distinction between absolute, intermediate and relative differential principal factors (DPF) ( [6] (6.3)) and about the values of the unit norm index ( [6] (1.3)) by actual computations.
All these conjectures have been proven by our most recent numerical investigations. Our classification is based on the Hasse-Iwasawa theorem about the Herbrand quotient of the unit group of the Galois closure N of L as a module over the relative group with respect to the cyclotomic subfield. It only involves the unit norm index and our 13 types of differential principal factorizations ( [6] Thm. 1.3), but not the index of subfield units ( [6] §5) in Parry’s class number formula ( [6] (5.1)).
We begin with a collection of explicit multiplicity formulas in §2 which are required for understanding the subsequent extensive presentation of our computational results in twenty tables of crucial invariants in §3. This information admits the classification of all 900 pure quintic fields with normalized radicands into 13 DPF types and the refined classification into similarity classes with representative prototypes in §4.
We draw the attention to remaining open questions in §3.3, and we collect theoretical consequences of our experimental results in §4.3. The exposition is concluded with a retrospective final §5.
2. Collection of Multiplicity Formulas
For the convenience of the reader, we provide a summary of formulas for calculating invariants of pure quintic fields with normalized fifth power free radicands and their associated pure metacyclic normal fields with a primitive fifth root of unity.
Let f be the class field theoretic conductor of the relatively quintic Kummer extension N/K over the cyclotomic field. It is also called the conductor of the pure quintic field L. The multiplicity of the conductor f indicates the number of non-isomorphic pure metacyclic fields N sharing the common conductor f, or also, according to ( [6] Prop. 2.1), the number of normalized fifth power free radicands whose fifth roots generate non-isomorphic pure quintic fields L sharing the common conductor f. The cardinality of a set S is denoted #S.
We adapt the general multiplicity formulas in ( [13] Thm. 2, p. 104) to the quintic case. If L is a field of species 1a ( [6] (2.6) and Exm. 2.2), i.e., then where. The explicit values of m in dependence on t are given in Table 1.
If L is a field of species 1b ( [6] (2.6) and Exm. 2.2), i.e., then where, and X j : = 1 5 ( 4 j − ( − 1 ) j ) , that is. The explicit values of m in dependence on u and v are given in Table 2.
If L is a field of species 2 ( [6] (2.6) and Exm. 2.2), i.e., then where, and
X j : = 1 5 ( 4 j − ( − 1 ) j ) , that is. The explicit values of m in dependence on u and v are given in Table 3.
3. Classification by DPF Types in 20 Numerical Tables3.1. DPF Types
The following twenty Tables 6-25 establish a complete classification of all 900 pure metacyclic fields with normalized radicands in the range. With the aid of PARI/GP [2] and MAGMA [5] we have determined the differential principal factorization type, T, of each field N by means of other invariants ( [6] Thm. 6.1). After several weeks of CPU time, the date of completion was September 17, 2018.
The possible DPF types are listed in dependence on in Table 4, where the symbol × in the column, resp., indicates the existence of a unit, resp., such that, resp.. The 5-valuation of the unit norm index is abbreviated by U ( [6] (1.3), (6.3)]. Here, η = 1 2 ( 1 + 5 ) denotes the fundamental unit of K + = Q ( 5 ) .
Multiplicity of fields of species 1a
t
0
1
2
3
4
5
m
1
4
16
64
256
1024
Multiplicity of fields of species 1b
u
v
0
1
2
3
4
5
0
m
0
1
3
13
51
205
1
0
4
12
52
204
820
2
0
16
48
208
816
3
0
64
192
832
4
0
256
768
Multiplicity of fields of species 2
u
v
0
1
2
3
4
5
0
m
0
0
1
3
13
51
1
1
0
4
12
52
204
2
4
0
16
48
208
816
3
16
0
64
192
832
4
64
0
256
768
Differential principal factorization types, T, of pure metacyclic fields N
T
U
A
I
R
2
−
−
1
0
2
2
−
−
1
1
1
2
−
−
1
2
0
2
−
−
2
0
1
2
−
−
2
1
0
2
−
−
3
0
0
1
×
−
1
0
1
1
×
−
1
1
0
1
×
−
2
0
0
1
−
×
1
0
1
1
−
×
1
1
0
1
−
×
2
0
0
0
×
×
1
0
0
3.2. Justification of the Computational Techniques
The steps of the following classification algorithm are ordered by increasing requirements of CPU time. To avoid unnecessary time consumption, the algorithm stops at early stages already, as soon as the DPF type is determined unambiguously. The illustrating subfield lattice of N is drawn in Figure 1.
Algorithm 3.1 (Classification into 13 DPF types.)
Input: a normalized fifth power free radicand.
Step 1: By purely rational methods, without any number field constructions, the prime factorization of the radicand D (including the counters, §4.2) is determined. If, , , then N is a Polya field of type; stop. If, or, then N is a Polya field of type; stop.
Step 2: The field L of degree 5 is constructed. The primes dividing the conductor f of N/K are determined, and their overlying prime ideals in L are computed. By means of at most principal ideal tests of
the elements of, the number , that is the cardinality of, is determined. If, then N is a Polya field. If, then N is of type; stop. If, , , then N is of type; stop. If, , then N is of type; stop.
Step 3: If or, then the field M of degree 10 is constructed. For the 2-split primes among the primes dividing the conductor f of N/K, the overlying prime ideals in M are computed. By means of at most principal ideal tests of the elements of , where for , the number, that is the cardinality of, is determined. If, then N is of type; stop. If, , then N is of type; stop.
Step 4: If, then the field N of degree 20 is constructed. For all 4-split primes among the primes dividing the conductor f of N/K, the overlying prime ideals in N are computed. By means of at most principal ideal tests of the elements of , where and for, the number, that is the cardinality of, is determined. If, then N is of type; stop. If, , then N is of type; stop. If, , then N is of type; stop.
Step 5: If the type of the field N is not yet determined uniquely, then and there remain the following possibilities. If, then N is of type, if, of type, if, and of type, if. If, then a fundamental system of units is constructed for the unit group of the field N of degree 20, and all relative norms of these units with respect to the cyclotomic subfield K are computed. If for some, , then N is of type, if, of type, if, and of type, if. Otherwise the conclusions are the same as for.
Output: the DPF type of the field and the decision about its Polya property.
Proof. The claims of Step 1 concerning the types are proved in items (1) and (2) of ( [6] Thm. 10.1).
For Step 2, the formulas (4.1) and (4.2) in ( [6] Thm. 4.1) give an -basis of the space of absolute differential factors, and the formulas (4.3) and (4.4) in ( [6] Cor. 4.1) determine bounds for the -dimension A of the space of absolute DPF in the field L of degree 5. The Polya property was characterized in ( [6] Thm. 10.5)], the claim concerning type follows from ( [6] Thm. 6.1), and the claims about the types from ( [6] Thm. 8.1 and Thm. 6.1).
For Step 3, the formulas (4.5) and (4.6) in ( [6] Thm. 4.3) give an -basis of the space of intermediate differential factors, and the formulas (4.7) and (4.8) in ( [6] Cor. 4.2) determine bounds for the -dimension I of the space of intermediate DPF in the field M of degree 10. The claims concerning the types are consequences of ( [6] Thm. 6.1),
For Step 4, the formulas (4.9) and (4.10) in ( [6] Thm. 4.4) give an -basis of the space of relative differential factors, and the formulas (4.11) and (4.12) in ( [6] Cor. 4.3) determine bounds for the -dimension R of the space of relative DPF in the field N of degree 20. The claims concerning the types are consequences of ( [6] Thm. 6.1).
Concerning Step 5, the signature of N is, whence the torsion free Dirichlet unit rank of N is given by. The claims about all types are consequences of ( [6] Thm. 6.1), including information on the constitution of the norm group.
Remark 3.1 Whereas the execution of Step 1 and 2 in Algorithm 3.1, implemented as a Magma program [5] , is a matter of a few seconds on a machine with two Intel XEON 8-core processors and clock frequency 2 GHz, the CPU time for Step 3 lies in the range of several minutes. The time requirement for Step 4 and 5 can reach hours or even days in spite of code optimizations for the calculation of units, in particular the use of the Magma procedures IndependentUnits() and SetOrderUnitsAreFundamental() prior to the call of UnitGroup().
3.3. Open Problems
We conjecture that considerable amounts of CPU time can be saved in our Algorithm 3.1 by computing the logarithmic 5-class numbers of the fields, which admit the determination of the logarithmic indices E, resp., of subfield units in the Parry [1] , resp. Kobayashi [14] [15] , class number relation, according to the formulas
However, first there would be required rigorous proofs of the heuristic connections between and the DPF types in Table 5, where implies type, implies type, implies type, but admits types, admits types, admits types, admits types, and admits types. seems to be impossible.
3.4. Conventions and Notation in the Tables
The normalized radicand of a pure metacyclic field N of degree 20 is minimal among the powers, , with corresponding exponents reduced modulo 5. The normalization of the radicands D provides a warranty that all fields are pairwise non-isomorphic ( [6] Prop. 2.1).
Prime factors are given for composite radicands D only. Dedekind’s species, S, of radicands is refined by distinguishing (species 1a) and (species 1b) among radicands (species 1). By the species and factorization of D, the shape of the conductor f is determined. We give the fourth power to avoid fractional exponents. Additionally, the multiplicity m indicates the number of non-isomorphic fields sharing a common conductor f (§2). The symbol briefly denotes the 5-valuation of the order of the class group of a number field F. By E we denote the exponent of the power in the index of subfield units.
Logarithmic indices <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x209.png" xlink:type="simple"/></inline-formula> of subfield units for DPF types, T
T
E
E+
or
E
E+
2
1
1
0
3
1
2
0
3
1
5
2
4
1
6
2
2
1
4
2
3
1
5
2
5
2
4
1
6
2
5
2
An asterisk denotes the smallest radicand with given Dedekind kind, DPF type and 5-class groups,. The latter are usually elementary abelian, except for the cases indicated by an additional asterisk (see §4.4).
Principal factors, P, are listed when their constitution is not a consequence of the other information. According to ( [6] Thm. 7.2., item (1)) it suffices to give the rational integer norm of absolute principal factors. For intermediate principal factors, we use the symbols with or with a prime element (which implies and thus also). Here, when a prime divides the radicand D. For relative principal factors, we use the symbols and with. Here, when a prime number divides the radicand D. (Kernel ideals in [6] §7)
The quartet indicates conditions which either enforce a reduction of possible DPF types or enable certain DPF types. The lack of a prime divisor together with the existence of a prime divisor and of D is indicated by a symbol × for the component 1. In these cases, only the two DPF types and can occur ( [6] Thm. 8.1).
A symbol × for the component 2 emphasizes a prime divisor of D and the possibility of intermediate principal factors in M, like and. A symbol × for the component 4 emphasizes a prime divisor of D and the possibility of relative principal factors in N, like and. The × symbol is replaced by if the facility is used completely, and by (×) if the facility is only used partially.
If D has only prime divisors or, a symbol × is placed in component 5. In these cases, can occur as a norm of some unit in. If it actually does, the × is replaced by ( [6] §8).
4. Statistical Evaluation and Refinements4.1. Statistics of DPF Types
The complete statistical evaluation of the following twenty Tables 6-25 is given in Table 26. The first ten columns show the absolute frequencies of pure metacyclic fields with various DPF types for the ranges with. The eleventh column lists the relative percentages of the five most frequent DPF types for the complete range of normalized radicands.
Among our 13 differential principal factorization types, type with 3-dimensional absolute principal factorization, , is clearly dominating with more than one third (36%) of all occurrences in the complete range, followed by type with 2-dimensional absolute principal factorization, , which covers nearly one quarter (23%) of all cases. The third place (nearly 18%) is occupied by type with mixed absolute and intermediate principal factorization, ,.
40 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x269.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
1
*2
1b
1
0
0
0
5
2
3
1b
1
0
0
0
5
3
*5
1a
1
0
0
0
5
4
*6
1b
3
0
0
1
6
5
*7
2
1
0
0
0
5
6
*10
1a
4
0
0
0
5
7
*11
1b
1
1
1
2
3
8
12
1b
3
0
0
1
6
9
13
1b
1
0
0
0
5
10
*14
1b
4
0
0
1
6
11
15
1a
4
0
0
0
5
12
17
1b
1
0
0
0
5
13
*18
2
1
0
0
0
5
14
*19
1b
1
1
1
2
3
15
20
1a
4
0
0
0
5
16
21
1b
4
0
0
1
6
17
*22
1b
3
1
1
3
4
18
23
1b
1
0
0
0
5
19
26
2
1
0
0
0
5
20
28
1b
4
0
0
1
6
21
29
1b
1
1
1
2
3
22
*30
1a
16
0
0
1
6
23
*31
1b
1
2
3
5
2
24
*33
1b
3
2
2
4
1
25
34
1b
3
0
0
1
6
26
*35
1a
4
0
0
1
6
27
37
1b
1
0
0
0
5
28
*38
1b
3
1
1
3
4
29
39
1b
3
0
0
1
6
30
40
1a
4
0
0
0
5
31
41
1b
1
1
1
2
3
32
*42
1b
12
1
2
5
6
33
43
2
1
0
0
0
5
34
44
1b
3
1
1
3
4
35
45
1a
4
0
0
0
5
36
46
1b
3
0
0
1
6
37
47
1b
1
0
0
0
5
38
48
1b
3
0
0
1
6
39
51
2
1
0
0
0
5
40
52
1b
3
0
0
1
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x426.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
(1,2,4,5)
T
P
41
53
1b
1
0
0
0
2
42
*55
1a
4
1
1
2
3
43
56
1b
4
0
0
1
6
44
*57
2
1
1
1
2
3
45
58
1b
3
1
1
3
4
46
59
1b
1
1
1
2
3
47
60
1a
16
0
0
1
6
48
61
1b
1
1
1
2
3
49
62
1b
3
1
1
3
4
50
63
1b
4
0
0
1
6
51
65
1a
4
0
0
0
5
52
*66
1b
13
1
2
5
6
53
67
1b
1
0
0
0
5
54
68
2
1
0
0
0
5
55
69
1b
3
0
0
1
6
56
*70
1a
16
0
0
1
6
57
71
1b
1
1
1
2
3
58
73
1b
1
0
0
0
5
59
74
2
1
0
0
0
5
60
75
1a
4
0
0
0
5
61
76
2
1
1
1
2
3
62
*77
1b
4
1
1
3
4
63
*78
1b
13
1
2
5
6
64
79
1b
1
1
1
2
3
65
80
1a
4
0
0
0
5
66
*82
2
1
1
1
2
3
67
83
1b
1
0
0
0
5
68
84
1b
12
1
2
5
6
69
85
1a
4
0
0
0
5
70
86
1b
4
0
0
1
6
71
87
1b
3
1
1
3
4
72
88
1b
3
2
2
4
1
73
89
1b
1
1
1
2
3
74
90
1a
16
0
0
1
6
75
91
1b
4
0
0
1
6
76
92
1b
3
0
0
1
6
77
93
2
1
1
1
2
3
78
94
1b
3
0
0
1
6
79
*95
1a
4
1
1
2
3
80
97
1b
1
0
0
0
5
81
99
2
1
1
1
2
3
82
*101
2
1
1
2
4
5
83
102
1b
13
1
2
5
6
84
103
1b
1
0
0
0
5
85
104
1b
3
0
0
1
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x617.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
86
105
1a
16
0
0
1
6
87
106
1b
3
0
0
1
6
88
107
2
1
0
0
0
5
89
109
1b
1
1
1
2
3
90
*110
1a
16
1
1
3
4
91
111
1b
3
0
0
1
6
92
112
1b
4
0
0
1
6
93
113
1b
1
0
0
0
5
94
*114
1b
13
1
2
5
6
95
115
1a
4
0
0
0
5
96
116
1b
3
1
1
3
4
97
117
1b
3
0
0
1
6
98
118
2
1
1
1
2
3
99
119
1b
4
0
0
1
6
100
120
1a
16
0
0
1
6
101
122
1b
3
1
1
3
4
102
*123
1b
3
2
3
6
3
103
124
2
1
1
1
2
3
104
*126
2
4
0
0
1
6
105
127
1b
1
0
0
0
5
106
129
1b
4
0
0
1
6
107
130
1a
16
0
0
1
6
108
*131
1b
1
2
2
4
1
109
*132
2
3
1
1
3
4
110
*133
1b
4
1
1
3
4
111
134
1b
3
0
0
1
6
112
136
1b
3
0
0
1
6
113
137
1b
1
0
0
0
5
114
138
1b
13
1
2
5
6
115
*139
1b
1
1
1
3
4
116
*140
1a
16
1
2
4
5
7
117
*141
1b
3
1
2
4
5
118
142
1b
3
1
1
3
4
119
143
2
1
1
1
2
3
120
145
1a
4
1
1
2
3
121
146
1b
3
0
0
1
6
122
147
1b
4
0
0
1
6
123
148
1b
3
0
0
1
6
124
*149
2
1
1
1
2
3
125
150
1a
16
0
0
1
6
126
*151
2
1
1
1
2
3
127
152
1b
3
1
1
3
4
128
153
1b
3
0
0
1
6
129
*154
1b
12
2
3
7
4
130
*155
1a
4
2
3
5
2
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x812.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
131
156
1b
13
1
2
5
6
132
157
2
1
0
0
0
5
133
158
1b
3
1
1
3
4
134
159
1b
3
1
2
4
5
5
135
161
1b
4
0
0
1
6
136
163
1b
1
0
0
0
5
137
164
1b
3
1
1
3
4
138
165
1a
16
1
1
3
4
139
166
1b
3
1
2
4
5
5
140
167
1b
1
0
0
0
5
141
168
2
4
0
0
1
6
142
170
1a
16
0
0
1
6
143
*171
1b
3
1
2
4
5
144
172
1b
4
0
0
1
6
145
173
1b
1
0
0
0
5
146
*174
2
3
1
1
3
4
147
175
1a
4
0
0
1
6
148
176
2
1
1
1
2
3
149
177
1b
3
1
1
3
4
150
178
1b
3
1
2
4
5
151
179
1b
1
1
1
2
3
152
*180
1a
16
1
2
4
5
3
153
181
1b
1
2
2
4
1
154
*182
2
4
1
2
4
5
7
155
183
1b
3
1
1
3
4
156
184
1b
3
0
0
1
6
157
185
1a
4
0
0
0
5
158
*186
1b
13
2
3
6
3
159
187
1b
3
1
1
3
4
160
188
1b
3
0
0
1
6
161
*190
1a
16
1
1
3
4
162
*191
1b
1
1
2
4
5
163
193
2
1
0
0
0
5
164
194
1b
3
1
2
4
5
165
195
1a
16
0
0
1
6
166
197
1b
1
0
0
0
5
167
198
1b
13
1
2
5
6
168
199
2
1
1
1
2
3
169
201
2
1
0
0
0
5
170
*202
1b
4
1
1
3
4
171
*203
1b
4
1
2
4
5
172
204
1b
13
1
2
5
6
173
205
1a
4
1
1
2
3
174
206
1b
3
1
2
4
5
175
207
2
1
0
0
0
5
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x1008.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
176
208
1b
3
0
0
1
6
177
*209
1b
3
2
3
7
4
178
*210
1a
64
1
2
5
6
179
*211
1b
1
3
5
9
2
180
212
1b
3
0
0
1
6
181
213
1b
3
1
1
3
4
182
214
1b
4
0
0
1
6
183
215
1a
4
0
0
1
6
184
217
1b
4
1
1
3
4
185
*218
2
1
1
2
4
5
2
186
219
1b
3
0
0
1
6
187
220
1a
16
1
1
3
4
188
221
1b
3
0
0
1
6
189
222
1b
13
1
2
5
6
190
223
1b
1
0
0
0
5
191
226
2
1
0
0
0
5
192
227
1b
1
0
0
0
5
193
228
1b
13
1
2
5
6
194
229
1b
1
1
1
2
3
195
230
1a
16
0
0
1
6
196
*231
1b
12
1
2
2
6
197
232
2
1
1
1
2
3
198
233
1b
1
0
0
0
5
199
234
1b
13
1
2
5
6
200
235
1a
4
0
0
0
5
201
236
1b
3
1
1
3
4
202
237
1b
3
1
1
3
4
203
238
1b
12
1
2
5
6
204
239
1b
1
1
1
2
3
205
240
1a
16
1
2
4
5
3
206
241
1b
1
1
1
2
3
207
244
1b
3
2
2
4
1
208
245
1a
4
0
0
1
6
209
246
1b
13
1
2
5
6
210
*247
1b
3
1
2
5
6
211
248
1b
3
1
1
3
4
212
249
2
1
0
0
0
5
213
251
2
1
1
1
2
3
214
252
1b
12
1
2
5
6
215
*253
1b
3
1
2
4
5
216
254
1b
3
0
0
1
6
217
255
1a
16
0
0
1
6
218
257
2
1
0
0
0
5
219
258
1b
12
1
2
5
6
220
*259
1b
4
1
2
5*
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x1205.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
221
260
1a
16
0
0
1
6
222
261
1b
3
1
1
3
4
223
262
1b
3
1
1
3
4
224
263
1b
1
0
0
0
5
225
264
1b
13
1
2
5
6
226
265
1a
4
0
0
0
5
227
*266
1b
12
1
2
5
6
228
267
1b
3
1
1
3
4
229
268
2
1
0
0
0
5
230
269
1b
1
1
1
2
3
231
270
1a
16
0
0
1
6
232
271
1b
1
1
2
4
5
233
272
1b
3
0
0
1
6
234
*273
1b
12
2
4
8
5
235
274
2
1
0
0
0
5
236
*275
1a
4
2
2
4
1
237
*276
2
3
0
0
1
6
238
277
1b
1
0
0
0
5
239
278
1b
3
1
1
3
4
240
279
1b
3
1
1
3
4
241
280
1a
16
0
0
1
6
242
*281
1b
1
3
5*
9*
2
243
282
2
3
0
0
1
6
244
283
1b
1
0
0
0
5
245
284
1b
3
1
1
3
4
246
*285
1a
16
1
2
5
6
247
*286
1b
13
2
3
7
4
248
*287
1b
4
2
2
4
1
249
*290
1a
16
1
2
4
5
29
250
291
1b
3
0
0
1
6
251
292
1b
3
0
0
1
6
252
293
2
1
0
0
0
5
253
294
1b
12
1
2
5
6
254
295
1a
4
1
1
2
3
255
296
1b
3
0
0
1
6
256
297
1b
3
1
1
3
4
257
*298
1b
4
1
2
4
5
5
258
299
2
1
0
0
0
5
259
*301
2
4
0
0
1
6
260
*302
1b
4
1
2
4
5
261
303
1b
4
1
1
3
4
262
304
1b
3
1
1
3
4
263
305
1a
4
2
2
4
1
264
306
1b
13
1
2
5
6
265
307
2
1
0
0
0
5
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x1402.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
266
308
1b
12
1
2
5
6
267
309
1b
3
1
2
4
5
268
310
1a
16
1
1
3
4
269
311
1b
1
1
1
2
3
270
312
1b
13
1
2
5
6
271
313
1b
1
0
0
0
5
272
314
1b
4
0
0
1
6
273
315
1a
16
1
2
4
5
7
274
316
1b
3
1
1
3
4
275
317
1b
1
0
0
0
5
276
318
2
3
0
0
1
6
277
*319
1b
3
2
2
5
2
278
321
1b
4
0
0
1
6
279
322
1b
12
2
4
8
5
280
323
1b
3
1
1
3
4
281
325
1a
4
0
0
0
5
282
326
2
1
0
0
0
5
283
327
1b
3
1
2
5
6
284
328
1b
3
1
1
3
4
285
*329
1b
4
1
2
4
5
47
286
*330
1a
64
2
4
9
6
287
331
1b
1
1
1
2
3
288
332
2
1
0
0
0
5
289
333
1b
3
0
0
1
6
290
334
1b
3
0
0
1
6
291
335
1a
4
0
0
0
5
292
336
1b
12
2
4
8
5
293
337
1b
1
0
0
0
5
294
339
1b
3
0
0
1
6
295
340
1a
16
0
0
1
6
296
*341
1b
3
3
5
9
2
297
342
1b
13
1
2
5
6
298
344
1b
4
0
0
1
6
299
345
1a
16
0
0
1
6
300
346
1b
3
1
2
4
5
301
347
1b
1
0
0
0
5
302
*348
1b
13
2
4
8
5
303
349
2
1
1
1
2
3
304
350
1a
16
0
0
1
6
305
351
2
1
0
0
0
5
306
353
1b
1
0
0
0
5
307
354
1b
13
1
2
5
6
308
355
1a
4
1
1
2
3
309
356
1b
3
1
2
4
5
310
357
2
4
0
0
1
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x1597.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
311
358
1b
3
1
1
3
4
312
359
1b
1
1
1
3
4
313
360
1a
16
0
0
1
6
314
362
1b
3
1
1
3
4
315
364
1b
12
1
2
5
6
316
365
1a
4
0
0
0
5
317
366
1b
13
2
3
6
3
318
367
1b
1
0
0
0
5
319
368
2
1
0
0
0
5
320
369
1b
3
2
3
6
3
321
370
1a
16
0
0
1
6
322
371
1b
4
0
0
1
6
323
372
1b
13
1
2
5
6
324
373
1b
1
0
0
0
5
325
374
2
3
1
1
3
4
326
376
2
1
0
0
0
5
327
*377
1b
3
2
3
6
3
328
378
1b
12
1
2
5
6
329
*379
1b
1
1
2
4
5
5
330
380
1a
16
1
1
3
4
331
381
1b
3
0
0
1
6
332
382
2
1
1
1
2
3
333
383
1b
1
0
0
0
5
334
*385
1a
16
1
1
3
4
335
386
1b
4
1
2
4
5
2
336
387
1b
4
0
0
1
6
337
388
1b
3
1
2
4
5
338
389
1b
1
1
1
2
3
339
*390
1a
64
1
2
5
6
2, 5
340
391
1b
3
0
0
1
6
341
393
2
1
1
1
2
3
342
394
1b
3
0
0
1
6
343
395
1a
4
1
1
2
3
344
396
1b
13
2
3
6
3
345
397
1b
1
0
0
0
5
346
*398
1b
4
1
1
3
4
347
*399
2
4
1
1
3
4
348
*401
2
1
2
3
5
2
349
402
1b
13
1
2
5
6
350
403
1b
3
1
1
3
4
351
404
1b
4
1
1
3
4
352
405
1a
4
0
0
0
5
353
406
1b
12
1
2
5
6
354
407
2
1
1
1
2
3
355
408
1b
13
1
2
5
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x1798.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
356
409
1b
1
1
1
2
3
357
410
1a
16
1
1
3
4
358
411
1b
3
0
0
1
6
359
412
1b
3
1
2
4
5
360
413
1b
4
1
1
3
4
361
414
1b
13
1
2
5
6
2, 3
362
415
1a
4
0
0
0
5
363
417
1b
3
1
1
3
4
364
*418
2
3
2
3
6
3
365
419
1b
1
1
1
3
4
366
420
1a
64
1
2
5
6
367
*421
1b
1
1
2
3
4
368
*422
1b
3
1
2
4
5
369
423
1b
3
1
2
4
5
370
424
2
1
0
0
0
5
371
425
1a
4
0
0
0
5
372
426
2
3
1
1
3
4
373
427
1b
4
1
1
3
4
374
428
1b
4
0
0
1
6
375
429
1b
13
2
3
7
4
376
430
1a
16
1
2
4
5
43
377
431
1b
1
1
1
2
3
378
433
1b
1
0
0
0
5
379
434
1b
12
1
2
5
6
380
435
1a
16
1
1
3
4
381
436
1b
3
1
1
3
4
382
437
1b
3
1
1
3
4
383
438
1b
13
1
2
5
6
3, 5
384
439
1b
1
1
1
2
3
385
440
1a
16
1
1
3
4
386
442
1b
13
1
2
5
6
387
443
2
1
0
0
0
5
388
444
1b
13
1
2
5
6
389
445
1a
4
1
1
2
3
390
446
1b
3
0
0
1
6
391
447
1b
4
1
1
3
4
392
449
2
1
1
1
2
3
393
*451
2
1
2
3
6
3
K ( 11 ) K ( 41 ) 4 , K ( 11 ) ,1 K ( 41 ) ,2 3
394
452
1b
3
0
0
1
6
395
453
1b
4
1
1
3
4
396
454
1b
3
0
0
1
6
397
455
1a
16
0
0
1
6
398
456
1b
13
1
2
5
6
399
457
2
1
0
0
0
5
400
458
1b
3
1
1
3
4
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2001.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
401
459
1b
3
0
0
1
6
402
460
1a
16
0
0
1
6
403
461
1b
1
1
2
3
4
404
*462
1b
52
2
4
9
6
5 2 × 7, 3 × 5 × 11 2
405
463
1b
1
0
0
0
5
406
464
1b
3
1
1
3
4
407
*465
1a
16
2
3
7*
4
408
466
1b
3
0
0
1
6
409
467
1b
1
0
0
0
5
410
468
2
3
0
0
1
6
411
469
1b
4
0
0
1
6
412
470
1a
16
0
0
1
6
413
471
1b
4
0
0
1
6
414
472
1b
3
1
1
3
4
415
*473
1b
4
2
3
7*
4
416
474
2
3
1
1
3
4
417
475
1a
4
1
1
2
3
418
476
2
4
0
0
1
6
419
477
1b
3
1
2
4
5
420
478
1b
3
1
1
3
4
421
479
1b
1
1
1
2
3
422
481
1b
3
0
0
1
6
423
*482
2
1
1
2
4
5
424
483
1b
12
1
2
5
6
425
485
1a
4
0
0
0
5
426
487
1b
1
0
0
0
5
427
488
1b
3
1
1
3
4
428
489
1b
3
1
2
4
5
429
490
1a
16
0
0
1
6
430
491
1b
1
1
1
2
3
431
492
1b
13
1
2
5
6
432
493
2
1
1
1
2
3
433
494
1b
13
1
2
5
6
434
495
1a
16
1
1
3
4
435
496
1b
3
1
1
3
4
436
497
1b
4
1
1
3
4
437
498
1b
13
1
2
5
6
438
499
2
1
1
1
2
3
439
501
2
1
0
0
0
5
440
*502
1b
4
2*
4*
8*
5
441
503
1b
1
0
0
0
5
442
504
1b
15
1
2
5
6
443
*505
1a
4
1
1
3
4
444
506
1b
13
1
2
5
6
445
508
1b
3
0
0
1
6
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2203.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
446
509
1b
1
1
1
2
3
447
510
1a
64
1
2
5
6
448
511
1b
4
0
0
1
6
449
513
1b
3
1
2
4
5
450
514
1b
4
0
0
1
6
451
515
1a
4
0
0
0
5
452
516
1b
12
1
2
5
6
453
*517
1b
3
2
3
6
3
454
518
2
4
0
0
1
6
455
519
1b
3
0
0
1
6
456
520
1a
16
1
2
4
5
2
457
521
1b
1
1
2
3
4
458
522
1b
13
1
2
5
6
459
523
1b
1
0
0
0
5
460
524
2
1
1
1
2
3
461
525
1a
16
0
0
1
6
462
526
2
1
0
0
0
5
463
527
1b
3
1
1
3
4
464
528
1b
13
1
2
5
6
465
530
1a
16
0
0
1
6
466
531
1b
3
1
1
3
4
467
*532
2
4
1
2
4
5
468
533
1b
3
1
1
3
4
469
534
1b
13
1
2
5
6
470
535
1a
4
0
0
1
6
471
536
1b
3
0
0
1
6
472
537
1b
3
1
1
3
4
473
538
1b
3
1
1
3
4
474
539
1b
4
1
1
3
4
475
540
1a
16
0
0
1
6
476
541
1b
1
2
2
4
1
477
542
1b
3
1
1
3
4
478
543
2
1
1
1
2
3
479
545
1a
4
1
1
2
3
480
*546
1b
52
2
4
9
6
481
547
1b
1
0
0
0
5
482
548
1b
3
0
0
1
6
483
549
2
3
2
2
4
1
484
*550
1a
16
2
3
6
3
485
*551
2
1
2
2
5
2
486
552
1b
13
1
2
5
6
487
553
1b
4
1
2
4
5
5
488
554
1b
3
0
0
1
6
489
555
1a
16
0
0
1
6
490
556
1b
3
1
1
3
4
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2407.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
491
557
2
1
0
0
0
5
492
558
1b
13
1
2
5
6
493
559
1b
4
0
0
1
6
494
560
1a
16
0
0
1
6
495
561
1b
13
1
2
5
6
496
562
1b
3
1
2
4
5
497
563
1b
1
0
0
0
5
498
564
1b
13
1
2
5
6
499
565
1a
4
0
0
0
5
500
566
1b
3
0
0
1
6
501
567
1b
4
0
0
1
6
502
568
2
1
1
1
2
3
503
569
1b
1
1
1
2
3
504
*570
1a
64
1
2
5
6
505
571
1b
1
2
2
4
1
506
572
1b
13
2
3
6
3
507
573
1b
3
1
1
3
4
508
*574
2
4
1
1
3
4
509
575
1a
4
0
0
0
5
510
577
1b
1
0
0
0
5
511
579
1b
4
0
0
1
6
512
580
1a
16
1
1
3
4
513
581
1b
4
0
0
1
6
514
582
2
3
0
0
1
6
515
583
1b
3
1
1
3
4
516
584
1b
3
0
0
1
6
517
585
1a
16
0
0
1
6
518
586
1b
4
1
2
4
5
2
519
587
1b
1
0
0
0
5
520
589
1b
3
2
2
5
2
521
*590
1a
16
2
3
7*
4
522
591
1b
3
0
0
1
6
523
592
1b
3
0
0
1
6
524
593
2
1
0
0
0
5
525
594
1b
13
1
2
5
6
526
595
1a
16
0
0
1
6
527
596
1b
4
1
2
4
5
528
597
1b
4
1
1
3
4
529
598
1b
13
1
2
5
6
530
599
2
1
1
1
2
3
531
600
1a
16
0
0
1
6
532
601
2
1
1
1
2
3
533
*062
1b
16
1
2
5
6
2, 7
534
603
1b
3
0
0
1
6
535
604
1b
4
1
2
4
5
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2602.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
536
605
1a
4
1
1
2
3
537
*606
1b
12
1
2
5
6
538
607
2
1
0
0
0
5
539
*609
1b
12
2
3
7
4
540
610
1a
16
1
1
3
4
541
611
1b
3
0
0
1
6
542
612
1b
13
1
2
5
6
543
613
1b
1
0
0
0
5
544
614
1b
4
1
2
4
5
2
545
615
1a
16
1
1
3
4
546
616
1b
12
2
3
7
4
547
617
1b
1
0
0
0
5
548
618
2
3
0
0
1
6
549
619
1b
1
1
1
2
3
550
*620
1a
16
2
4*
8*
5
551
621
1b
3
0
0
1
6
552
622
1b
3
2
2
4
1
553
623
1b
4
1
1
3
4
554
624
2
3
0
0
1
6
555
626
2
1
0
0
0
5
556
*627
1b
13
3
4
9
2
557
628
1b
4
0
0
1
6
558
629
1b
3
0
0
1
6
559
630
1a
64
1
2
5
6
3, 5
560
631
1b
1
1
1
2
3
561
632
2
1
1
1
2
3
562
633
1b
3
1
2
4
5
563
634
1b
3
0
0
1
6
564
635
1a
4
0
0
0
5
565
636
1b
13
1
2
5
6
566
637
1b
4
0
0
1
6
567
*638
1b
13
2
3
7
4
568
639
1b
3
1
1
3
4
569
641
1b
1
1
2
4
5
570
642
1b
12
1
2
5
6
571
643
2
1
0
0
0
5
572
644
1b
12
1
2
5
6
573
645
1a
16
1
2
4
5
43
574
646
1b
13
1
2
5
6
575
647
1b
1
0
0
0
5
576
*649
2
1
2
2
5
2
577
650
1a
16
0
0
1
6
578
651
2
4
1
1
3
4
579
652
1b
3
1
2
4
5
5
580
653
1b
1
0
0
0
5
44 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2799.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
581
654
1b
13
1
2
5
6
582
655
1a
4
1
1
2
3
583
656
1b
3
2
2
4
1
584
657
2
1
0
0
0
5
585
658
1b
12
1
2
5
6
586
659
1b
1
1
1
2
3
587
*660
1a
64
1
2
5
6
588
661
1b
1
1
1
2
3
589
662
1b
3
1
1
3
4
590
663
1b
13
1
2
5
6
591
664
1b
3
1
2
4
5
592
*665
1a
16
1
1
3
4
593
666
1b
13
1
2
5
6
594
667
1b
3
1
1
3
4
595
668
2
1
0
0
0
5
596
669
1b
3
0
0
1
6
597
670
1a
16
1
2
4
5
2
598
*671
1b
3
3
4
8
1
599
673
1b
1
0
0
0
5
600
674
2
1
0
0
0
5
601
677
1b
1
0
0
0
5
602
678
1b
13
1
2
5
6
603
679
1b
4
0
0
1
6
604
680
1a
16
0
0
1
6
605
681
1b
3
0
0
1
6
606
*682
2
3
2
3
6
3
607
683
1b
1
0
0
0
5
608
684
1b
13
1
2
5
6
609
685
1a
4
0
0
0
5
610
687
1b
3
1
1
3
4
611
688
1b
4
0
0
1
6
612
689
1b
3
0
0
1
6
613
690
1a
64
1
2
5
6
614
*691
1b
1
3
4
8
1
615
692
1b
3
1
2
4
5
616
*693
2
4
2
3
6
3
617
694
1b
3
0
0
1
6
618
*695
1a
4
1
2
4
5
139
619
696
1b
13
1
2
5
6
620
697
1b
3
1
1
3
4
621
698
1b
4
1
1
3
4
622
699
2
1
0
0
0
5
623
700
1a
16
0
0
1
6
624
701
2
1
2
3
5
2
47 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x2996.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
625
*702
1b
13
2
4
8
5
626
703
1b
3
1
1
3
4
627
705
1a
16
0
0
1
6
628
706
1b
3
0
0
1
6
629
*707
2
4
1
1
3
4
630
708
1b
13
1
2
5
6
631
709
1b
1
1
1
2
3
632
*710
1a
16
2
2
4
1
633
711
1b
3
1
1
3
4
634
712
1b
3
1
2
4
5
635
713
1b
3
1
1
3
4
636
714
1b
52
2
4
9
6
637
715
1a
16
1
1
3
4
638
716
1b
3
1
1
3
4
639
717
1b
3
1
1
3
4
640
718
2
1
1
1
2
3
641
719
1b
1
1
1
2
3
642
720
1a
16
0
0
1
6
643
721
1b
4
0
0
1
6
644
723
1b
3
1
1
3
4
645
724
2
1
1
1
2
3
646
725
1a
4
1
1
2
3
647
726
2
3
1
1
3
4
648
727
1b
1
0
0
0
5
649
728
1b
12
1
2
5
6
5, 7
650
730
1a
16
0
0
1
6
651
731
1b
4
0
0
1
6
652
732
2
3
1
1
3
4
653
733
1b
1
0
0
0
5
654
734
1b
3
0
0
1
6
655
735
1a
16
0
0
1
6
656
737
1b
3
1
2
4
5
657
738
1b
13
1
2
5
6
658
739
1b
1
1
1
2
3
659
740
1a
16
1
2
4
5
37
660
741
1b
13
1
2
5
6
661
742
1b
12
1
2
5
6
662
743
2
1
0
0
0
5
663
744
1b
13
1
2
5
6
664
*745
1a
4
1
1
3
4
665
746
1b
3
1
2
4
5
666
747
1b
3
0
0
1
6
667
748
1b
13
1
2
5
6
668
*749
2
4
1
2
4
5
669
*751
2
1
2
2
4
1
670
752
1b
3
0
0
1
6
671
753
1b
4
1
1
3
4
44 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x3210.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
672
754
1b
13
1
2
5
6
673
755
1a
4
1
1
3
4
674
756
1b
12
1
2
5
6
675
757
2
1
0
0
0
5
676
758
1b
3
1
1
3
4
677
759
1b
13
2
3
6
3
678
760
1a
16
1
1
3
4
679
761
1b
1
2
3
5
2
680
762
1b
13
1
2
5
6
681
763
1b
4
1
1
3
4
682
764
1b
3
2
2
4
1
683
765
1a
16
0
0
1
6
684
766
1b
3
0
0
1
6
685
767
1b
3
1
1
3
4
686
769
1b
1
1
1
2
3
687
*770
1a
64
1
2
5
6
688
771
1b
4
0
0
1
6
689
772
1b
4
1
2
4
5
2
690
773
1b
1
0
0
0
5
691
774
2
4
0
0
1
6
692
775
1a
4
2
3
5
2
693
776
2
1
0
0
0
5
694
777
1b
12
1
2
5
6
695
778
1b
3
1
1
3
4
696
*779
1b
3
3
4
8
1
697
780
1a
64
1
2
5
6
698
781
1b
3
3
5
9
2
699
782
2
3
0
0
1
6
700
783
1b
3
1
1
3
4
701
*785
1a
4
1
2
4
5
702
786
1b
13
2
3
7
4
703
787
1b
1
0
0
0
5
704
788
1b
3
0
0
1
6
705
789
1b
3
0
0
1
6
706
790
1a
16
1
2
4
5
5
707
791
1b
4
0
0
1
6
708
792
1b
13
1
2
5
6
709
793
2
1
1
2
4
5
710
794
1b
3
0
0
1
6
711
795
1a
16
0
0
1
6
712
796
1b
4
1
1
3
4
713
797
1b
1
0
0
0
5
714
*798
1b
52
2
4
9
6
715
799
2
1
0
0
0
5
46 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x3407.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
716
801
2
1
1
1
2
3
717
802
1b
4
1
1
3
4
718
803
1b
3
1
1
3
4
719
804
1b
13
1
2
5
6
720
805
1a
16
0
0
1
6
721
806
1b
13
2
3
6
3
722
807
2
1
1
1
2
3
723
*808
1b
4
2
2
4
1
724
809
1b
1
1
1
2
3
725
810
1a
16
0
0
1
6
726
811
1b
1
1
1
2
3
727
812
1b
12
1
2
5
6
728
813
1b
3
1
1
3
4
729
814
1b
13
2
3
6
3
730
815
1a
4
0
0
0
5
731
816
1b
13
1
2
5
6
732
817
1b
4
1
1
3
4
733
818
2
1
1
1
2
3
734
819
1b
12
1
2
5
6
735
820
1a
16
2
2
4
1
736
821
1b
1
1
1
2
3
737
822
1b
13
2
4
8
5
738
823
1b
1
0
0
0
5
739
824
2
1
0
0
0
5
740
*825
1a
16
1
2
4
5
741
826
2
4
1
1
3
4
742
827
1b
1
0
0
0
5
743
828
1b
13
1
2
5
6
744
829
1b
1
1
1
3
4
745
830
1a
16
0
0
1
6
746
831
1b
3
1
2
4
5
747
833
1b
4
0
0
1
6
748
834
1b
13
1
2
5
6
749
835
1a
4
0
0
0
5
750
836
1b
13
2
3
7
4
751
837
1b
3
1
1
3
4
752
838
1b
3
1
1
3
4
753
839
1b
1
1
1
2
3
754
840
1a
64
1
2
5
6
755
842
1b
3
1
2
4
5
756
*843
2
1
1
2
3
4
757
844
1b
3
1
2
4
5
758
845
1a
4
0
0
0
5
759
846
1b
13
1
2
5
6
2, 3
760
847
1b
4
1
1
3
4
761
848
1b
3
0
0
1
6
47 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x3620.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
762
849
2
1
0
0
0
5
763
850
1a
16
0
0
1
6
764
851
2
1
0
0
0
5
765
852
1b
13
1
2
5
6
766
853
1b
1
0
0
0
5
767
854
1b
12
1
2
5
6
768
855
1a
16
1
1
3
4
769
856
1b
4
0
0
1
6
770
857
2
1
0
0
0
5
771
*858
1b
51
2
4
9
6
772
859
1b
1
1
1
2
3
773
860
1a
16
0
0
1
6
774
*861
1b
12
3
4
8
1
775
862
1b
3
1
2
4
5
776
863
1b
1
0
0
0
5
777
865
1a
4
0
0
0
5
778
866
1b
3
0
0
1
6
779
868
2
4
1
1
3
4
780
869
1b
3
2
2
5
2
781
870
1a
64
1
2
5
6
782
871
1b
3
0
0
1
6
783
872
1b
3
1
1
3
4
784
873
1b
3
0
0
1
6
785
874
2
3
1
1
3
4
786
876
2
3
0
0
1
6
787
877
1b
1
0
0
0
5
788
878
1b
3
1
1
3
4
789
879
1b
4
1
2
4
5
3
790
880
1a
16
2
2
4
1
791
881
1b
1
1
2
3
4
792
883
1b
1
0
0
0
5
793
884
1b
13
1
2
5
6
794
885
1a
16
1
1
3
4
795
886
1b
4
0
0
1
6
796
887
1b
1
0
0
0
5
797
888
1b
13
1
2
5
6
2, 3
798
889
1b
4
0
0
1
6
799
890
1a
16
1
1
3
4
800
891
1b
3
1
1
3
4
801
892
1b
3
0
0
1
6
802
*893
2
1
1
1
3
4
803
*894
1b
12
1
2
5
6
804
895
1a
4
1
1
2
3
805
897
1b
13
1
2
5
6
806
898
1b
4
1
1
3
4
807
899
2
1
2
2
5
2
808
901
2
1
0
0
0
5
47 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x3827.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
809
*902
1b
13
2
3
7
4
810
903
1b
16
1
2
5
6
811
904
1b
3
0
0
1
6
812
905
1a
4
1
1
2
3
813
906
1b
12
1
2
5
6
814
907
2
1
0
0
0
5
815
908
1b
3
0
0
1
6
816
909
1b
4
2
2
4
1
817
910
1a
64
1
2
5
6
818
911
1b
1
1
1
2
3
819
912
1b
13
1
2
5
6
820
913
1b
3
1
1
3
4
821
914
1b
4
1
2
4
5
2
822
915
1a
16
1
1
3
4
823
916
1b
3
1
1
3
4
824
917
1b
4
1
1
3
4
825
918
2
3
0
0
1
6
826
919
1b
1
1
1
2
3
827
920
1a
16
0
0
1
6
828
921
1b
4
0
0
1
6
829
922
1b
3
1
2
4
5
830
923
1b
3
2
2
4
1
831
*924
2
12
1
2
5
6
832
925
1a
4
0
0
0
5
833
926
2
1
0
0
0
5
834
927
1b
3
1
2
4
5
835
929
1b
1
1
1
2
3
836
930
1a
64
2
4
9
6
5, 31
837
931
1b
4
1
1
3
4
838
932
2
1
0
0
0
5
839
933
1b
3
1
1
3
4
840
934
1b
3
0
0
1
6
841
935
1a
16
2
2
4
1
842
936
1b
13
1
2
5
6
3, 5
843
937
1b
1
0
0
0
5
844
938
1b
12
1
2
5
6
845
939
1b
3
1
2
4
5
5
846
940
1a
16
0
0
1
6
847
941
1b
1
2
2
4
1
848
942
1b
12
2
4
8
5
157
849
943
2
1
1
1
2
3
850
944
1b
3
1
1
3
4
851
945
1a
16
0
0
1
6
852
946
1b
12
1
2
5
6
853
947
1b
1
0
0
0
5
854
948
1b
13
1
2
5
6
855
949
2
1
0
0
0
5
45 pure metacyclic fields with normalized radicands<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x4037.png" xlink:type="simple"/></inline-formula>
No.
D
Factors
S
f4
m
VL
VM
VN
E
T
P
856
950
1a
16
1
1
3
4
857
951
2
1
0
0
0
5
858
952
1b
12
1
2
5
6
859
953
1b
1
0
0
0
5
860
954
1b
13
1
2
5
6
861
*955
1a
4
2*
3*
6*
3
862
956
1b
3
1
1
3
4
863
*957
2
3
2
2
5
2
864
958
1b
3
1
2
4
5
865
959
1b
4
0
0
1
6
866
962
1b
13
1
2
5
6
867
963
1b
4
0
0
1
6
868
964
1b
3
1
2
4
5
869
965
1a
4
0
0
1
6
870
966
1b
52
2
4
9
6
871
967
1b
1
0
0
0
5
872
969
1b
13
1
2
5
6
873
970
1a
16
0
0
1
6
874
971
1b
1
2
2
4
1
875
973
1b
4
1
1
3
4
876
974
2
1
0
0
0
5
877
975
1a
16
0
0
1
6
878
976
2
1
1
1
2
3
879
977
1b
1
0
0
0
5
880
978
1b
13
1
2
5
6
881
979
1b
3
2
3
7
4
882
980
1a
16
1
2
4
5
7
883
981
1b
3
1
1
3
4
884
*982
2
1
1
1
2
3
885
983
1b
1
0
0
0
5
886
984
1b
13
1
2
5
6
887
985
1a
4
0
0
0
5
888
986
1b
13
2
4
8
5
889
987
1b
12
1
2
5
6
890
988
1b
13
1
2
5
6
891
989
1b
4
0
0
1
6
892
990
1a
64
1
2
5
6
893
991
1b
1
1
2
3
4
894
993
2
1
1
1
2
3
895
994
1b
12
2
3
7
4
896
995
1a
4
1
1
3
4
897
996
1b
13
1
2
5
6
898
997
1b
1
0
0
0
5
899
998
1b
4
1
1
3
4
900
999
2
1
0
0
0
5
Absolute frequencies of differential principal factorization types
Type
100
200
300
400
500
600
700
800
900
1000
%
1
2
3
4
5
5
5
9
9
9
10
17
23
30
35
42
52
57
63
75
8.3
0
0
0
1
1
3
5
5
7
8
0
2
4
7
8
11
15
18
22
23
7
24
40
54
80
94
108
126
146
161
17.9
25
55
88
117
148
187
222
259
290
324
36.0
0
0
1
1
3
4
4
4
6
7
8
14
19
23
31
35
38
44
51
53
5.9
26
45
67
95
110
128
150
165
184
208
23.4
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
4
4
5
1
2
4
5
5
6
6
6
6
7
3
6
8
9
11
13
15
17
18
19
Total
81
168
258
347
438
530
622
715
807
900
100.0
It is striking that type with 2-dimensional relative principal factorization, , and type with 2-dimensional intermediate principal factorization, , are populated rather sparsely, in favour of a remarkable contribution by type with mixed intermediate and relative principal factorization, (place four with 8%).
The appearance of the four types with norm representation, , of the primitive fifth root of unity is marginal ( [6] Thm. 8.2), in spite of the parametrized contribution by all prime conductors to type, as we shall prove in Theorem 4.1 (1) in §4.3.
4.2. Similarity Classes and Prototypes
In [7] , we came to the conviction that for deeper insight into the arithmetical structure of the fields under investigation, the prime factorization of the class field theoretic conductor f of the abelian extension N/K over the cyclotomic field and the primary invariants of all involved 5-class groups must be taken in consideration. These ideas were inspired by [16] [17] and have lead to the concept of similarity classes and representative prototypes, which refines the differential principal factorization (DPF) types
Let t be the number of primes distinct from 5 which divide the conductor f. Among these prime numbers, we separately count free primes, restrictive primes, 2-split primes, and 4-split primes. The multiplicity is given in terms of, according to §2, and the dimensions of various spaces of primitive ambiguous ideals over the finite field are given in terms of
, according to [6] §4. By we denote the fundamental unit
of. The dimensions of the spaces of absolute, intermediate and relative DPF over are denoted by A, I and R, identical with the additive (logarithmic) version in ( [6] Thm. 6.1). Further, let be the maximal real intermediate field of N/L, and denote by the subgroup of the unit group of generated by the units of all conjugate fields of and of, where is a primitive fifth root of unity. For a number field F, let be the 5-valuation of the class number of F.
Definition 4.1 A set of normalized fifth power free radicands is called a similarity class if the associated pure quintic fields share the following common multiplets of invariants:
• the refined Dedekind species, where
• the differential principal factorization type, where
• the structure of the 5-class groups, where
Warning 4.1 To reduce the number of invariants, we abstain from defining additional counters and for free splitting prime divisors of the conductor f. However, we point out that occasionally a similarity class in the sense of Definition 4.1 will be split in two separate classes, having the same invariants, but distinct contributions to the counters u and, resp.. For instance, the similarity classes [77] and [202] with and share identical multiplets of invariants (species 1b), (type), and. But and are due to 7, and are due to 11, in the former case, whereas and are due to 2, and are due to 101, in the latter case. Therefore, the contributions by primes congruent to will be indicated by writing and, resp..
We also emphasize that in the rare cases of non-elementary 5-class groups, the actual structures (abelian type invariants) of the 5-class groups will be taken into account, and not only the 5-valuations.
Definition 4.2 The minimal element of a similarity class (with respect to the natural order of positive integers) is called the representative prototype of the class, which is denoted by writing its prototype in square brackets.
The remaining elements of a similarity class, which are bigger than the prototype, only reproduce the arithmetical invariants of the prototype and do not provide any additional information, exept possibly about other primary components of the class groups, that is the structure of ℓ-class groups of the fields for.
Whereas there are only 13 DPF types of pure quintic fields, the number of similarity classes is obviously infinite, since firstly the number t of primes dividing the conductor is unbounded and secondly the number of states, defined by the triplet of 5-valuations of class numbers, is also unlimited.
Given a fixed refined Dedekind species, the set of all associated normalized fifth power free radicands D usually splits into several similarity classes defined by distinct DPF types (type splitting). Occasionally it even splits further into different structures of 5-class groups, called states, with increasing complexity of abelian type invariants (state splitting).
The 134 prototypes of pure quintic fields are listed in the Tables 27-30. By we denote the number of elements of the similarity class defined by the prototype, truncated at the upper bound of our systematic investigations.
4.3. General Theorems on DPF Types and Polya Fields
There is only a single finite similarity class [5] = {5}, characterized by the
46 prototypes <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x4331.png" xlink:type="simple"/></inline-formula> of pure metacyclic fields
No.
Factors
S
m
VL
VM
VN
E
T
P
1
2
1b
1
0
0
0
5
71
2
5
1a
1
0
0
0
5
1
3
6
1b
3
0
0
1
6
77
4
7
2
1
0
0
0
5
18
5
10
1a
4
0
0
0
5
31
6
11
1b
1
1
1
2
3
14
7
14
1b
4
0
0
1
6
44
8
18
2
1
0
0
0
5
37
9
19
1b
1
1
1
2
3
27
10
22
1b
3
1
1
3
4
35
11
30
1a
16
0
0
1
6
37
12
31
1b
1
2
3
5
2
2
13
33
1b
3
2
2
4
1
8
14
35
1a
4
0
0
1
6
6
15
38
1b
3
1
1
3
4
44
16
42
1b
12
1
2
5
6
22
17
55
1a
4
1
1
2
3
6
18
57
2
1
1
1
2
3
10
19
66
1b
13
1
2
5
6
17
20
70
1a
16
0
0
1
6
14
21
77
1b
4
1
1
3
4
7
22
78
1b
13
1
2
5
6
37
23
82
2
1
1
1
2
3
15
24
95
1a
4
1
1
2
3
9
25
101
2
1
1
2
4
5
1
26
110
1a
16
1
1
3
4
11
27
114
1b
13
1
2
5
6
20
28
123
1b
3
2
3
6
3
2
29
126
2
4
0
0
1
6
6
30
131
1b
1
2
2
4
1
6
31
132
2
3
1
1
3
4
5
32
133
1b
4
1
1
3
4
7
33
139
1b
1
1
1
3
4
4
34
140
1a
16
1
2
4
5
7
5
35
141
1b
3
1
2
4
5
19
36
149
2
1
1
1
2
3
6
37
151
2
1
1
1
2
3
3
38
154
1b
12
2
3
7
4
3
39
155
1a
4
2
3
5
2
2
40
171
1b
3
1
2
4
5
6
41
174
2
3
1
1
3
4
3
42
180
1a
16
1
2
4
5
3
5
43
182
2
4
1
2
4
5
7
1
44
186
1b
13
2
3
6
3
7
45
190
1a
16
1
1
3
4
9
46
191
1b
1
1
2
4
5
3
44 prototypes <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x4540.png" xlink:type="simple"/></inline-formula> of pure metacyclic fields
No.
Factors
S
m
VL
VM
VN
E
T
P
47
202
1b
4
1
1
3
4
6
48
203
1b
4
1
2
4
5
2
49
209
1b
3
2
3
7
4
2
50
210
1a
64
1
2
5
6
5
51
211
1b
1
3
5
9
2
1
52
218
2
1
1
2
4
5
2
1
53
231
1b
12
1
2
5
6
5
54
247
1b
3
1
2
5
6
2
55
253
1b
3
1
2
4
5
4
56
259
1b
4
1
2
5*
6
1
57
266
1b
12
1
2
5
6
3
58
273
1b
12
2
4
8
5
4
59
275
1a
4
2
2
4
1
2
60
276
2
3
0
0
1
6
10
61
281
1b
1
3
5*
9*
2
1
62
285
1a
16
1
2
5
6
1
63
286
1b
13
2
3
7
4
3
64
287
1b
4
2
2
4
1
1
65
290
1a
16
1
2
4
5
29
2
66
298
1b
4
1
2
4
5
5
2
67
301
2
4
0
0
1
6
1
68
302
1b
4
1
2
4
5
2
69
319
1b
3
2
2
5
2
3
70
329
1b
4
1
2
4
5
47
7
71
330
1a
64
2
4
9
6
2
72
341
1b
3
3
5
9
2
2
73
348
1b
13
2
4
8
5
2
74
377
1b
3
2
3
6
3
1
75
379
1b
1
1
2
4
5
5
1
76
385
1a
16
1
1
3
4
1
77
390
1a
64
1
2
5
6
4
78
398
1b
4
1
1
3
4
7
79
399
2
4
1
1
3
4
2
80
401
2
1
2
3
5
2
2
81
418
2
3
2
3
6
3
1
82
421
1b
1
1
2
3
4
5
83
422
1b
3
1
2
4
5
6
84
451
2
1
2
3
6
3
1
85
462
1b
52
2
4
9
6
1
86
465
1a
16
2
3
7*
4
1
87
473
1b
4
2
3
7*
4
1
88
482
2
1
1
2
4
5
2
89
502
1b
4
*2
*4
*8
5
1
90
505
1a
4
1
1
3
4
2
39 prototypes <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x4750.png" xlink:type="simple"/></inline-formula> of pure metacyclic fields
No.
Factors
S
m
VL
VM
VN
E
T
P
91
517
1b
3
2
3
6
3
1
92
532
2
4
1
2
4
5
1
93
546
1b
52
2
4
9
6
3
94
550
1a
16
2
3
6
3
1
95
551
2
1
2
2
5
2
1
96
570
1a
64
1
2
5
6
2
97
574
2
4
1
1
3
4
3
98
590
1a
16
2
3
*7
4
1
99
602
1b
16
1
2
5
6
2
100
606
1b
12
1
2
5
6
2
101
609
1b
12
2
3
7
4
1
102
620
1a
16
2
4*
8*
5
1
103
627
1b
13
3
4
9
2
1
104
638
1b
13
2
3
7
4
2
105
649
2
1
2
2
5
2
2
106
660
1a
64
1
2
5
6
2
107
665
1a
16
1
1
3
4
1
108
671
1b
3
3
4
8
1
1
109
682
2
3
2
3
6
3
1
110
691
1b
1
3
4
8
1
1
111
693
2
4
2
3
6
3
1
112
695
1a
4
1
2
4
5
139
1
113
702
1b
13
2
4
8
5
2
114
707
2
4
1
1
3
4
1
115
710
1a
16
2
2
4
1
4
116
745
1a
4
1
1
3
4
2
117
749
2
4
1
2
4
5
1
118
751
2
1
2
2
4
1
1
119
770
1a
64
1
2
5
6
1
120
779
1b
3
3
4
8
1
1
121
785
1a
4
1
2
4
5
1
122
798
1b
52
2
4
9
6
1
123
808
1b
4
2
2
4
1
2
124
825
1a
16
1
2
4
5
1
125
843
2
1
1
2
3
4
1
126
858
1b
51
2
4
9
6
1
127
861
1b
13
3
4
8
1
1
128
893
2
1
1
1
3
4
1
129
894
1b
12
1
2
5
6
1
exceptional number of primes dividing the conductor f (here). The invariants of this unique metacyclic Polya field N are given by
[5], species 1a, ,
type, , and. (4.4)
We conjecture that all the other similarity classes are infinite. Precisely four of them can actually be given by parametrized infinite sequences in a deterministic way aside from the intrinsic probabilistic nature of the occurrence of primes in
5 prototypes 900 < M < 1000 of pure metacyclic fields
No.
M
Factors
S
f 4
m
VL
VM
VN
E
( 1,2,4,5 )
T
P
| M |
130
902
2 × 11 × 41
1b
5 2 2 4 11 4 41 4
13
2
3
7
4
( − , − , ( × ) , − )
β 2
11 × 5, K ( 11 ) × K ( 41 )
1
131
924
2 2 × 3 × 7 × 11
2
2 4 3 4 7 4 11 4
12
1
2
5
6
( − , − , × , − )
γ
3 × 7,7 × 11
1
132
955
5 × 191
1a
5 6 191 4
4
2*
3*
6*
3
( − , − , ⊗ , − )
α 2
K , K 1
1
133
957
3 × 11 × 29
2
3 4 11 4 29 4
3
2
2
5
2
( − , ⊗ , ( × ) , − )
α 3
K ( 11 ) , K ( 29 )
1
134
982
2 × 491
2
2 4 491 4
1
1
1
2
3
( − , − , ⊗ , − )
α 2
K , K 1
2
residue classes and of composite integers with assigned shape of prime decomposition. This was proved in ( [6] Thm. 10.1)and ( [7] Thm. 2.1).
Theorem 4.1 Each of the following infinite sequences of conductors unambiguously determines the DPF type of the pure metacyclic fields N in the associated multiplet with members.
1) with, gives rise to a singulet, , with DPF type,
2) with, , gives rise to a singulet, , with DPF type,
3) with, , gives rise to a quartet, , with homogeneous DPF type,
4) with, , gives rise to a singulet, , with DPF type.
In fact, the shape of the conductors in Theorem 4.1 does not only determine the refined Dedekind species and the DPF type, but also the structure of the 5-class groups of the fields L, M and N.
Corollary 4.1 The invariants of the similarity classes defined by the four infinite sequences of conductors in Theorem 4.1 are given as follows, in the same order:
[7], species 2, ,
type, ( U , η , ζ ; A , I , R ) = ( 0 , × , × ; 1 , 0 , 0 ) , and; (4.5)
[2], species 1b, ,
type, , and; (4.6)
[10], species 1a, ,
type, , and; (4.7)
[18], species 2, ,
type, , and. (4.8)
The pure metacyclic fields N associated with these four similarity classes are Polya fields.
Remark 4.1 The statements concerning 5-class groups in Corollary 4.1 were proved by Parry in ( [1] Thm. IV, p. 481), where Formula (10) gives the shape of radicands associated with the conductors in our Theorem 4.1.
Proof. (of Theorem 4.1 and Corollary 4.1) It only remains to show the claims for the composite radicands associated with conductors and. See ( [6] Thm. 10.6).
For similarity classes distinct from the four infinite classes in Theorem 4.1 we cannot provide deterministic criteria for the DPF type and for the homogeneity of multiplets with. In general, the members of a multiplet belong to distinct similarity classes, thus giving rise to heterogeneous DPF types. We explain these phenomena with the simplest cases where only two or three DPF types are involved (type splitting).
Theorem 4.2 Each of the following infinite sequences of conductors admits precisely three DPF types of the pure metacyclic fields N in the associated quartet with members.
1) with, gives rise to a quartet with possibly heterogeneous DPF type, , conjecturally always,
2) with, gives rise to a quartet with possibly heterogeneous DPF type, , conjecturally always.
Example 4.1 It is quite easy to find complete quartets, whose members are spread rather widely. The smallest quartet belonging to the first infinite sequence contains the member outside of the range of our systematic computations. We have determined its DPF type separately and thus discovered a homogeneous quartet of type. However, we cannot generally exclude the occurrence of heterogeneous quartets.
Corollary 4.2 The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.2 are given as follows, in the same order. The statements concerning 5-class groups are only conjectural. Each sequence splits in two similarity classes.
The classes for are:
[35], species 1a, ,
type, , and; (4.9)
[785], species 1a, ,
type, , and. (4.10)
The classes for are:
[301], species 2, ,
type, , and; (4.11)
[749], species 2, ,
type, , and. (4.12)
All pure metacyclic fields N associated with these four similarity classes are Polya fields.
Proof. (of Theorem 4.2 and Corollary 4.2) We use, and ( [6] Thm. 6.1).
Remark 4.2. The statements on 5-class groups in Corollary 4.2 have been verified for all examples with by our computations. In particular, the occurrence of the radicands and, both with, proves the impossibility of the general claim for the two situations mentioned in ( [18] Lem. 3.3 (ii) and (iv), p. 204) and ( [15] Thm. 5 (ii) and (iv), p. 5), partially also indicated in ( [1] Thm. IV (11), p. 481).
Theorem 4.3 Each of the following infinite sequences of conductors admits precisely two DPF types of the pure metacyclic fields N in the associated hexadecuplet with members.
1) with, , both gives rise to a hexadecuplet with possibly heterogeneous DPF type, ,
2) with, , only one gives rise to a hexadecuplet with possibly heterogeneous DPF type,.
Example 4.2 It is not difficult to find complete hexadecuplets, whose members are spread rather widely. The smallest hexadecuplet
belonging to the first infinite sequence contains the members outside of the range of our systematic computations. We have determined their DPF type separately and thus discovered a heterogeneous hexadecuplet (in the same order) of type
Corollary 4.3 The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.3 are given as follows, in the same order. The statements concerning 5-class groups are only conjectural. Each sequence splits into two similarity classes.
The classes for, both are:
[30], species 1a, ,
type, , and; (4.13)
[180], species 1a, ,
type, , and. (4.14)
The classes for, only one are:
[70], species 1a, ,
type, , and; (4.15)
[140], species 1a, ,
type, , and. (4.16)
Only the pure metacyclic fields N of type associated with (14) and (16) are Polya fields.
Proof. (of Theorem 4.3 and Corollary 4.3) See ( [6] Thm. 10.7).
Theorem 4.4 A pure metacyclic field with prime radicand has a prime conductor, and possesses the Polya property, regardless of its DPF type and the complexity of its 5-class group structure.
Proof. This is an immediate consequence of ( [6] Thm. 10.5 and Thm. 6.1), taking into account that we have the value for the number of primes dividing the conductor in the present situation, and thus the estimate in ( [6] Cor. 4.1) yields. For the Polya property we must have, according to ( [6] Thm. 10.5), which admits the DPF types or ( [6] Thm. 1.3 and Tbl. 1). However, DPF type is excluded by ( [6] Cor. 4.2), since the requirement cannot be fulfilled in our situation where either and for or and for.
Theorem 4.5 A pure metacyclic field with prime radicand but has a composite conductor, and the following conditions are equivalent:
1) N possesses the Polya property.
2).
3) The prime ideal with is principal.
4) N is of DPF type either or or.
Proof. This is a consequence of ( [6] Thm. 10.5 and Thm. 6.1), taking into account that the prime 5 is not included in the current definition of the counter t (with value in the present situation), and thus the estimate in ( [6] Cor. 4.1) must be replaced by. For the Polya property we must have ( [6] Thm. 10.5), which determines the DPF types or ( [6] Thm. 1.3 and Tbl. 1). However, DPF type is excluded by the prime dividing the conductor (( [6] Thm. 8.1)).
Inspired by the last two theorems, it is worth ones while to summarize, for each kind of prime radicands, what is known about the possibilities for differential principal factorizations.
Theorem 4.6 Let be a pure metacyclic field with prime radicand.
1) If with or, then N is of type.
2) If with, then N is of one of the types.
3) If with, then N is of one of the types.
4) If with but, then N is of type.
5) If with but, then N is of one of the types.
6) If with but, then N is of one of the types.
A pure metacyclic field with prime radicand can never be of any of the types.
Proof. By making use of the bounds [6] §4 for -dimensions of spaces of differential principal factors (DPF),
we can determine the possible DPF types of pure metacyclic fields with prime radicands. We start with a few general observations.
Firstly, if, resp., is prime, then N is of Dedekind species 2, resp. 1a, with prime power conductor, resp., and, whence and the types with are forbidden. However, if and is prime, then the congruence requirement eliminates the types, the field N is of Dedekind species 1b with composite conductor, and, whence and type with is discouraged. So, the types and are generally forbidden for prime radicands.
Secondly, for a prime radicand which splits in M, the space of radicals is a 1-dimensional subspace of absolute DPF contained in the 2-dimensional space of differential factors generated by the two prime ideals of M over D. Consequently, in this special situation there arises an additional constraint for the dimension of the space of intermediate DPF, which must be contained in the 1-dimensional complement. This generally excludes type with for prime radicands.
1) If with, then, , and thus,. These conditions eliminate the types with either or or, and only type remains admissible.
2) If with, then, , and thus, , , whence the types with either or or are excluded, and only the types remain admissible.
3) If with, then, , and thus, , , whence the types with either or are excluded, and only the types remain admissible.
4) If with but, then, , and thus,. These conditions eliminate the types with either or or, and only the types remain admissible. However, the congruence requirement modulo 25 discourages the types, and only type is possible.
5) If with but, then, , , and thus, , , whence the types with either or or are forbidden. The types are excluded by congruence conditions, and only the types remain admissible.
6) If with but then, , , and thus, , , whence the types with either or are forbidden. The types are excluded by congruence conditions, and only the types remain admissible.
Example 4.3 Concerning numerical realizations of Theorem 4.6, we refer to Corollary 4.1 for the parametrized infinite sequences [7] and [2] which realize item (1) and (4). (See also Tables 23 and 19 for the types and.) In all the other cases, there occurs type splitting:
The similarity class [149] partially realizes item (2). (See Table 38 for the type.) Outside the range of our systematic investigations, we found that the similarity class [1049] realizes type. Realizations of the type are unknown up to now.
The similarity classes [401], [151] and [101] partially realize item (3). (See Table 31, Table 32 and Table 40 for the types, and.) Outside the range of our systematic investigations, we found that the similarity class [1151], resp. [3251], realizes type, resp.. Realizations of the types and are unknown up to now.
The similarity classes [139], [19] and [379] completely realize item (5). (See Table 35, Table 38 and Table 39 for the types, and.)
The similarity classes [31], [11], [191] and [211] partially realize item (6). (See Table 31, Table 32, Table 34 and Table 37 for the types, , and.) Realizations of the types, and are unknown up to now.
4.4. Non-Elementary 5-Class Groups
Although most of the 5-class groups of pure metacyclic fields N, maximal real subfields M and pure quintic subfields L are elementary abelian, there occur sparse examples with non-elementary structure. For instance, we have only 8 occurrences within the range of our computations:
1), for (type),
2), , for prime (type),
3), for (type),
4), for (type),
5), , , for (type),
6), for (type),
7), , for (type),
8), , , for (type).
However, outside the range of systematic computations, we additionally found:
a), , , for prime (type),
b), , for prime (type),
c), , , for prime (type),
d), , , for prime (type).
We point out that in all of the last four examples, the normal field N is a Polya field, since the radicands D are primes, the conductors are primes, and thus all primitive ambiguous ideals are principal, generated by the radical and its powers. Consequently, there seems to be no upper bound for the complexity of 5-class groups of pure metacyclic Polya fields N in Theorem 4.4.
4.5. Refinement of DPF Types by Similarity Classes
Based on the definition of similarity classes and prototypes in §4.2, on the explicit listing of all prototypes in the range between 2 and 103 in Tables 27-30, and on theoretical foundations in §4.3, we are now in the position to establish the intended refinement of our 13 differential principal factorization types into similarity classes in Tables 31-43, as far as the range of our computations for normalized radicands is concerned. The cardinalities refine the statistical evaluation in Table 26.
DPF types are characterized by the multiplet, refined Dedekind species, S, by the multiplet, and 5-class groups by the multiplet.
DPF type splits into 3 similarity classes in the ground state and 2 similarity classes in the first excited state. Summing up the partial frequencies 6+3 of these states in Table 31 yields the modest absolute frequency 9 of type in the range, as given in Table 26. The logarithmic subfield unit index of type is restricted to the single value. Type is the unique type with 2-dimensional relative principal factorization,.
The logarithmic subfield unit index of DPF type can take two values, either or. Type with splits into 5 similarity classes in the ground state and 6 similarity classes in the first excited state. Type with splits into 7 similarity classes in the ground state and 4 similarity classes in the first excited state. Summing up the partial frequencies 40 + 7, resp. 24 + 4, of these states in Table 32 yields the considerable absolute frequency 75 of type in the range, as given in Table 26. Type is the unique type with mixed intermediate and relative principal factorization,.
DPF type splits into 4 similarity classes in the ground state and 1 similarity class in the first excited state. Summing up the partial frequencies 7 + 1 of these states in Table 33 yields the modest absolute frequency 8 of type in the range, as given in Table 26. The logarithmic subfield unit index of type is restricted to the unique value. Type is the unique type with 2-dimensional intermediate principal factorization,.
The logarithmic subfield unit index of DPF type can take two values, either or. Type with consists of 3 similarity classes in the ground state. Type with splits into 5 similarity classes in the ground state and 2 similarity classes in the first excited state. Summing up the partial frequencies 9, resp. 12 + 2, of these states in Table 34 yields the modest absolute frequency 23 of type in the range, as given in Table 26. Type is the unique type with mixed absolute and relative principal factorization, and.
DPF type splits into 16 similarity classes in the ground state and 9 similarity classes in the first excited state. Summing up the partial frequencies 146 + 15 of these states in Table 35 yields the high absolute frequency 161 of type in the range, as given in Table 26. The logarithmic subfield unit index of type is restricted to the unique value. Type is the unique type with mixed absolute and intermediate principal factorization, and.
DPF type splits into 6 similarity classes in the ground state , 16 similarity classes in the first excited state , and 5 similarity classes in the second excited state . Summing up the partial frequencies 188+128+8 of these states in Table 36 yields the maximal absolute frequency 324 of type among all 13 types in the range , as given in Table 26. The logarithmic subfield subfield unit index of type is restricted to the unique value . Type is the unique type with 3-dimensional absolute principal factorization,. However, the 1-dimensional subspace is formed by radicals, and only the complementary 2-dimensional subspace is non-trivial.
The logarithmic subfield unit index of DPF type can take two values, either or. Type with splits into 2 similarity classes in the ground state. Type with consists of 1 similarity class in the ground state. Summing up the partial frequencies 6+1 of these states in Table 37 yields the modest absolute frequency 7 of type in the range, as given in Table 26. Type is a type with 1-dimensional relative principal factorization,.
DPF type splits into 4 similarity classes in the ground state and 1 similarity class in the first excited state. Summing up the partial frequencies 52+1 of these states in Table 38 yields the considerable absolute frequency 53 of type in the range, as given in Table 26. The logarithmic subfield unit index of type is restricted to the unique value. Type is a type with 1-dimensional intermediate principal factorization,.
DPF type splits into 3 similarity classes in the ground state, 16 similarity classes in the first excited state, and 3 similarity classes in the second excited state. Summing up the partial frequencies 139+61+8 of these states in Table 39 yields the high absolute frequency 208 of type in the range, as given in Table 26. The logarithmic subfield unit index of type is restricted to the unique value. Type is a type with 2-dimensional absolute principal factorization,.
The logarithmic subfield unit index of DPF type is restricted to the unique value. Type consists of 1 similarity class in the ground state. The frequency 1 of this state in Table 40 coincides with the negligible absolute frequency 1 of type in the range, as given in Table 26. Type is a type with 1-dimensional relative principal factorization,.
DPF type consists of 3 similarity classes. The modest absolute frequency 5 of type in the range, given in Table 26, is the sum 2+1+2 of partial frequencies in Table 41. Type only occurs with logarithmic subfield
unit index. It is a type with 1-dimensional intermediate principal factorization,.
DPF type splits in 2 similarity classes, [35] and [301]. The modest absolute frequency 7 of type in the range, given in Table 26, is the sum 6+1 of partial frequencies in Table 42. Type only occurs with logarithmic subfield unit index. It is a type with 2-dimensional absolute principal factorization,. However, it should be pointed out that outside of the range of our systematic investigations we found an excited state for the similarity class [1505], where has three prime divisors, additionally to the ground state .
Splitting of type <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5499.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5499.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5500.png" xlink:type="simple"/></inline-formula>: 3 similarity classes
No splitting of type<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5505.png" xlink:type="simple"/></inline-formula>,<inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5505.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5506.png" xlink:type="simple"/></inline-formula>: 1 similarity class
Splitting of type η , ( U , η , ζ ; A , I , R ) = ( 1, − , × ; 2,0,0 ) : 2 similarity classes
No.
S
e0
t
u
v
m
n
s2
s4
VL
VM
VN
E
M
|M|
1
1a
6
1
1
0
4
1
0
0
0
0
1
6
35
6
2
2
0
2
2
0
4
2
0
0
0
0
1
6
301
1
DPF type splits into the unique finite similarity class [5] with only a single element and the infinite parametrized sequence [7] consisting of all prime radicands congruent to . The small absolute frequency 19 of type in the range , given in Table 26, is the sum in Table 43. Since no theoretical argument disables the occurrence of type for composite radicands D with prime factors 5 and, we conjecture that such cases will appear in bigger ranges with. Type only occurs with logarithmic subfield unit index, and is the unique type where every unit of K occurs as norm of a unit of N, that is.
4.6. Increasing Dominance of DPF Type <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5523.png" xlink:type="simple"/></inline-formula> for <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5523.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5524.png" xlink:type="simple"/></inline-formula>
In this final section, we want to show that the careful book keeping of similarity classes with representative prototypes in Tables 31-43 is useful for the quantitative illumination of many other phenomena. For an explanation, we select the phenomenon of absolute principal factorizations.
The statistical distribution of DPF types in Table 26 has proved that type with 324 occurrences, that is 36%, among all 900 fields with normalized radicands in the range is doubtlessly the high champion of all DPF types. This means that there is a clear trend towards the maximal possible extent of 3-dimensional spaces of absolute principal factorizations, , in spite of the disadvantage that the estimate in the formulas (4.3) and (4.4) of ( [6] Cor. 4.1) prohibits type for conductors f with prime divisors.
For the following investigation, we have to recall that the number T of all prime factors of is given by for fields of Dedekind’s species 1, where, and by for fields of Dedekind’s species 2, where.
Conductors f with prime factors occur in six tables,
Splitting of type <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5538.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5538.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="//html.scirp.org/file/4-5301607x5539.png" xlink:type="simple"/></inline-formula>: 2 similarity classes
No.
S
e0
t
u
v
m
n
s2
s4
VL
VM
VN
E
M
|M|
1
1a
6
0
0
0
1
0
0
0
0
0
0
5
5
1
2
2
0
1
1
0
1
1
0
0
0
0
0
5
7
18
1 case of type in a single similarity class of Table 32,
1 case of type in a single similarity class of Table 33,
7 cases of type in a single similarity class of Table 34,
10 cases of type in 5 similarity classes of Table 35,
126 cases of type in 16 similarity classes of Table 36,
8 cases of type in 3 similarity classes of Table 39,
that is, a total of 153 cases, with respect to the complete range of our computations. Consequently, we have an increase of type from 36.0%,
with respect to the entire database, to, with respect to.
The feature is even aggravated for conductors f with prime factors, which exclusively occur in Table 36. There are 4 similarity classes with, namely [462], [546], [798], [858], with a total of 6 elements, all (100%) with associated fields of type.
5. Conclusions
In this paper, it was our intention to realize Parry’s suggestion ( [1] p. 484) concerning a numerical investigation of pure quintic number fields. For this purpose, we first developed theoretical foundations in a series of preparatory papers [6] [7] [8] [9] which expand the original germs in [12] . Since the non-Galois fields L do not contain the full wealth of arithmetical structures, we had to consider their pure metacyclic normal closures of degree 20.
On the one hand, this enabled us to use the Galois cohomology of the unit group UN with respect to the relative automorphism group over the cyclotomic field for defining 13 exhaustive and mutually exclusive differential principal factorization (DPF) types (Table 4 in §3.1), based on the unit norm index, where, which is connected with the order of the group of primitive ambiguous principal ideals via the Takagi/Hasse/Iwasawa-Theorem on the Herbrand quotient of UN, and on a natural decomposition into the dimensions of the -vector spaces of absolute, intermediate and relative differential principal factors ( [6] , eqn. (6.3)).
On the other hand, our theory of the relatively cyclic quintic Kummer extension N/K as a 5-ring class field modulo the conductor over K admitted the calculation of the multiplicity of the conductor f (§2), which is the number of non-isomorphic pure metacyclic fields N sharing the common conductor f and forming a multiplet [8] .
Equipped with this theoretical background, we were able to develop our Classification Algorithm 3.1 in §3.2, and to prove that it determines the DPF type of L and N in finitely many steps. It also decides whether the normal field N is a Polya field or not. (It is known that L is a (trivial!) Polya field if and only if it possesses class number [6] .)
The algorithm was implemented as a Magma program script [3] [4] [5] and applied to the 900 fields N with normalized fifth power free radicands in the range, after some preliminary experiments with Pari/GP [2] . The result is documented in the twenty Tables 6-25 of §3.4. It is in perfect accordance with our theoretical predictions in 1991 [12] . Actually, each type occurs indeed, the most hardboiled type not earlier than for the radicand, and it is interesting to study the statistical distribution of the types in §4.1, Table 6. Concerning the reliability of our extensive database, and for understanding the degree of precision contained in our paper, we point out that all tables have been thoroughly double checked with respect to misprints and copy-paste errors for at least three times.
Nevertheless, after the completion of the statistics in §4, we came to the conviction that for deeper insight into the arithmetical structure of pure metacyclic fields N, the prime factorization of the class field theoretic conductor f of the abelian extension N/K (invariants) and the primary invariants of all involved 5-class groups, partially given by the 5-valuations, should be taken in consideration. In the last section, we present a corresponding refinement of the DPF types in similarity classes (§4.2), which is very useful for various applications. It reduces the number 900 of investigated radicands to 134 representative prototypes (§4.5). The remaining 766 radicands, which are bigger than the prototype of their similarity class, only reproduce the arithmetical invariants of the prototype and do not provide any additional relevant information. The rare phenomenon of non-elementary 5-class groups is documented in §4.4. Theorems on DPF-types of members of multiplets are proved in §4.3, and the striking dominance of type with 3-dimensional (maximal) absolute principal factorization for radicands with three and more prime divisors is presented and discussed in §4.6.
Acknowledgements
We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): project P 26008-N25. This work is dedicated to the memory of Charles J. Parry († 25 December 2010) who suggested a numerical investigation of pure quintic number fields. We are indebted to the anonymous referees for valuable suggestions concerning an improvement of the paper’s layout.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Mayer, D.C. (2019) Tables of Pure Quintic Fields. Advances in Pure Mathematics, 9, 347-403. https://doi.org/10.4236/apm.2019.94017
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and their Galois closures 
with 3 possible types to pure quintic fields 
and their pure metacyclic normal fields 
with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D<103
. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K)
of N over the cyclotomic field K=Q(ξ5)
, by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN))
by the number #(PN/K/PK)
of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K
. The precise structure of the
F5
-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (U<SUB>N</SUB>:U<SUB>0</SUB>).
The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.