Pure Quintic Fields 3

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Pure Quintic Fields 3 Previous Page § 7. DPF types of pure quintic fields. We define thirteen possible differential principal factorization types of pure quintic number fields L = Q(D1/5), according to the generators of the group NormN|kUN as the primary invariant and the triplet (a,i,r) consisting of the order a = #Pa of the group of absolute DPF of L|Q, the index i = (Pi:Pa) in the group of intermediate DPF of M|k+, and the index r = (Pr:Pi) in the group of relative DPF of N|k, as the secondary invariant (similar to but not identical with our definitions in [Ma2]). The connection between the various quantities is given by the chain of equations a * i * r = #Pr = (PNGal(N|k):Pk) = (EN|k:UN1 - σ) = 5 (Uk:NormN|kUN). Note that the unit index (UN:U0) does not enter the definition of the DPF types. NormN|kUN (Uk:NormN|kUN) (PNGal(N|k):Pk) a i r DPF type <-1,η5> 25 125 5 1 25 α1 <-1,η5> 25 125 5 5 5 α2 <-1,η5> 25 125 5 25 1 α3 <-1,η5> 25 125 25 1 5 β1 <-1,η5> 25 125 25 5 1 β2 <-1,η5> 25 125 125 1 1 γ <-1,η> 5 25 5 1 5 δ1 <-1,η> 5 25 5 5 1 δ2 <-1,η> 5 25 25 1 1 ε <-1,ζ,η5> 5 25 5 1 5 ζ1 <-1,ζ,η5> 5 25 5 5 1 ζ2 <-1,ζ,η5> 5 25 25 1 1 η <-1,ζ,η> 1 5 5 1 1 θ § 8. Computational results. With the aid of PARI/GP and MAGMA we have determined the DPF type and other invariants of the 90 pure quintic number fields L = Q(D1/5) with normalized radicands D = q1e1 … qses (minimal among the powers Dn, 1 ≤ n ≤ 4, with corresponding ejs reduced mod 5) in the range 2 ≤ D ≤ 110 (date of completion: Jan 11th, 2014). Prime factors are given for composite D only (qj ≡ + 1 (mod 5) in red, qj ≡ - 1 (mod 5) in magenta, qj ≡ ± 7 (mod 25) in green). The first kind of fields is refined by distinguishing 5|D (kind Ia) and (5,D) = 1 (kind Ib). The exponent of the power in the unit index (UN:U0) = 5e is denoted by e. The unit norm index (Uk:NormN|kUN) is abbreviated by u. The symbol Cl briefly denotes the 5-class group Cl5 of a number field. An asterisk (*) denotes the smallest radicand with given type and 5-class groups. No. D factors kind Cl(L) Cl(M) Cl(N) e u DPF type * 1 2 Ib 1 1 1 5 5 ε * 2 3 Ib 1 1 1 5 5 ε 3 5 Ia 1 1 1 5 1 θ * 4 6 2,3 Ib 1 1 (5) 6 25 γ * 5 7 II 1 1 1 5 1 θ 6 10 2,5 Ia 1 1 1 5 5 ε 7 11 Ib (5) (5) (5,5) 3 25 α2 * 8 12 3,22 Ib 1 1 (5) 6 25 γ 9 13 Ib 1 1 1 5 5 ε 10 14 2,7 Ib 1 1 (5) 6 25 γ 11 15 3,5 Ia 1 1 1 5 5 ε 12 17 Ib 1 1 1 5 5 ε 13 18 2,32 II 1 1 1 5 5 ε 14 19 Ib (5) (5) (5,5) 3 5 δ2 * 15 20 5,22 Ia 1 1 1 5 5 ε 16 21 3,7 Ib 1 1 (5) 6 25 γ 17 22 2,11 Ib (5) (5) (5,5,5) 4 25 β2 * 18 23 Ib 1 1 1 5 5 ε 19 26 2,13 II 1 1 1 5 5 ε 20 28 7,22 Ib 1 1 (5) 6 25 γ 21 29 Ib (5) (5) (5,5) 3 5 δ2 22 30 2,3,5 Ia 1 1 (5) 6 25 γ 23 31 Ib (5,5) (5,5,5) (5,5,5,5,5) 2 25 α1 * 24 33 3,11 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2 * 25 34 2,17 Ib 1 1 (5) 6 25 γ 26 35 5,7 Ia 1 1 (5) 6 5 η * 27 37 Ib 1 1 1 5 5 ε 28 38 2,19 Ib (5) (5) (5,5,5) 4 25 β2 29 39 3,13 Ib 1 1 (5) 6 25 γ 30 40 5,23 Ia 1 1 1 5 5 ε 31 41 Ib (5) (5) (5,5) 3 25 α2 32 42 2,3,7 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ * 33 43 II 1 1 1 5 1 θ 34 44 11,22 Ib (5) (5) (5,5,5) 4 25 β2 35 45 5,32 Ia 1 1 1 5 5 ε 36 46 2,23 Ib 1 1 (5) 6 25 γ 37 47 Ib 1 1 1 5 5 ε 38 48 3,24 Ib 1 1 (5) 6 25 γ 39 51 3,17 II 1 1 1 5 5 ε 40 52 13,22 Ib 1 1 (5) 6 25 γ 41 53 Ib 1 1 1 5 5 ε 42 55 5,11 Ia (5) (5) (5,5) 3 25 α2 43 56 7,23 Ib 1 1 (5) 6 25 γ 44 57 3,19 II (5) (5) (5,5) 3 5 δ2 45 58 2,29 Ib (5) (5) (5,5,5) 4 25 β2 46 59 Ib (5) (5) (5,5) 3 5 δ2 47 60 3,5,22 Ia 1 1 (5) 6 25 γ 48 61 Ib (5) (5) (5,5) 3 25 α2 49 62 2,31 Ib (5) (5) (5,5,5) 4 25 β2 50 63 7,32 Ib 1 1 (5) 6 25 γ 51 65 5,13 Ia 1 1 1 5 5 ε 52 66 2,3,11 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 53 67 Ib 1 1 1 5 5 ε 54 68 17,22 II 1 1 1 5 5 ε 55 69 3,23 Ib 1 1 (5) 6 25 γ 56 70 2,5,7 Ia 1 1 (5) 6 25 γ 57 71 Ib (5) (5) (5,5) 3 25 α2 58 73 Ib 1 1 1 5 5 ε 59 74 2,37 II 1 1 1 5 5 ε 60 75 3,52 Ia 1 1 1 5 5 ε 61 76 19,22 II (5) (5) (5,5) 3 5 δ2 62 77 7,11 Ib (5) (5) (5,5,5) 4 25 β2 63 78 2,3,13 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 64 79 Ib (5) (5) (5,5) 3 5 δ2 65 80 5,24 Ia 1 1 1 5 5 ε 66 82 2,41 II (5) (5) (5,5) 3 25 α2 67 83 Ib 1 1 1 5 5 ε 68 84 3,7,22 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 69 85 5,17 Ia 1 1 1 5 5 ε 70 86 2,43 Ib 1 1 (5) 6 25 γ 71 87 3,29 Ib (5) (5) (5,5,5) 4 25 β2 72 88 11,23 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2 73 89 Ib (5) (5) (5,5) 3 5 δ2 74 90 2,5,32 Ia 1 1 (5) 6 25 γ 75 91 7,13 Ib 1 1 (5) 6 25 γ 76 92 23,22 Ib 1 1 (5) 6 25 γ 77 93 3,31 II (5) (5) (5,5) 3 25 α2 78 94 2,47 Ib 1 1 (5) 6 25 γ 79 95 5,19 Ia (5) (5) (5,5) 3 5 δ2 80 97 Ib 1 1 1 5 5 ε 81 99 11,32 II (5) (5) (5,5) 3 25 α2 82 101 II (5) (5,5) (5,5,5,5) 5 5 ζ1 * 83 102 2,3,17 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 84 103 Ib 1 1 1 5 5 ε 85 104 13,23 Ib 1 1 (5) 6 25 γ 86 105 3,5,7 Ia 1 1 (5) 6 25 γ 87 106 2,53 Ib 1 1 (5) 6 25 γ 88 107 II 1 1 1 5 1 θ 89 109 Ib (5) (5) (5,5) 3 5 δ2 90 110 2,5,11 Ia (5) (5) (5,5,5) 4 25 β2 On Jan 13th, 2014, we additionally determined the DPF type and other invariants of the 35 pure quintic number fields L = Q(D1/5) with normalized radicands in the range 111 ≤ D ≤ 150. No. D factors kind Cl(L) Cl(M) Cl(N) e u DPF type * 91 111 3,37 Ib 1 1 (5) 6 25 γ 92 112 7,24 Ib 1 1 (5) 6 25 γ 93 113 Ib 1 1 1 5 5 ε 94 114 2,3,19 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 95 115 5,23 Ia 1 1 1 5 5 ε 96 116 29,22 Ib (5) (5) (5,5,5) 4 25 β2 97 117 13,32 Ib 1 1 (5) 6 25 γ 98 118 2,59 II (5) (5) (5,5) 3 5 δ2 99 119 7,17 Ib 1 1 (5) 6 25 γ 100 120 3,5,23 Ia 1 1 (5) 6 25 γ 101 122 2,61 Ib (5) (5) (5,5,5) 4 25 β2 102 123 3,41 Ib (5,5) (5,5,5) (5,5,5,5,5,5) 3 25 α2 * 103 124 31,22 II (5) (5) (5,5) 3 25 α2 104 126 2,7,32 II 1 1 (5) 6 25 γ 105 127 Ib 1 1 1 5 5 ε 106 129 3,43 Ib 1 1 (5) 6 25 γ 107 130 2,5,13 Ia 1 1 (5) 6 25 γ 108 131 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2 109 132 3,11,22 II (5) (5) (5,5,5) 4 25 β2 110 133 7,19 Ib (5) (5) (5,5,5) 4 25 β2 111 134 2,67 Ib 1 1 (5) 6 25 γ 112 136 17,23 Ib 1 1 (5) 6 25 γ 113 137 Ib 1 1 1 5 5 ε 114 138 2,3,23 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ 115 139 Ib (5) (5) (5,5,5) 4 25 β2 116 140 5,7,22 Ia (5) (5,5) (5,5,5,5) 5 5 ε * 117 141 3,47 Ib (5) (5,5) (5,5,5,5) 5 5 ε 118 142 2,71 Ib (5) (5) (5,5,5) 4 25 β2 119 143 11,13 II (5) (5) (5,5) 3 25 α2 120 145 5,29 Ia (5) (5) (5,5) 3 5 δ2 121 146 2,73 Ib 1 1 (5) 6 25 γ 122 147 3,72 Ib 1 1 (5) 6 25 γ 123 148 37,22 Ib 1 1 (5) 6 25 γ 124 149 II (5) (5) (5,5) 3 5 δ2 125 150 2,3,52 Ia 1 1 (5) 6 25 γ Summarized, we finally obtain the following Statistics of DPF types, expressed by absolute frequencies and relative frequencies in percent (%). Arrows indicate monotonic tendencies (↑ … increasing, ↓ … descending). D ≤ tot ε γ α1 α2 β2 δ2 θ η ζ1 Cl(N)=1 Cl(N)=(5) 50 38 14 11 1 3 3 2 3 1 0 17 11 37 29 8 8 5 8 44 29 100 81 26 25 1 10 7 8 3 1 0 29 22 32 31 12 9 10 4 36 27 150 125 33 45 1 14 14 12 4 1 1 35 39 26↓ 36↑ 11 11↑ 10 3↓ 28↓ 31 The missing types have been found by us later in 2018: type β1 occurs for D = 186 = 2*3*31, type δ1 occurs for D = 211 (prime), type α3 occurs for D = 319 = 11*29, and type ζ2 did not appear until D = 505 = 5*101. Next Page

Trade, Science, Art and Industry Principal investigator of the International Research Project Towers of p-Class Fields over Algebraic Number Fields supported by the Austrian Science Fund (FWF): P 26008-N25

Bibliographical References: [MAGMA] The MAGMA Group, MAGMA Computational Algebra System, Version 2.21-1, Sydney, 2014, http://magma.maths.usyd.edu.au [Ma2] D. C. Mayer, Classification of dihedral fields, University of Manitoba, Winnipeg, Manitoba, Canada, 1991. [PARI] The PARI Group, PARI/GP, Version 2.7.2 Bordeaux, 2014, http://pari.math.u-bordeaux.fr

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