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Principalization algorithm via class group structure. (English. French summary) Zbl 1361.11073

Summary: For an algebraic number field \(K\) with \(3\)-class group \(\mathrm{Cl}_3(K)\) of type \((3, 3)\), the structure of the \(3\)-class groups \(\mathrm{Cl}_3(N_i)\) of the four unramified cyclic cubic extension fields \(N_i\), \(1 \leq i \leq 4\), of \(K\) is calculated with the aid of presentations for the metabelian Galois group \(\mathrm{G}_3^2(K) = \mathrm{Gal}(\mathrm{F}_3^2(K)|K)\) of the second Hilbert \(3\)-class field \(\mathrm{F}_3^2(K)\) of \(K\). In the case of a quadratic base field \(K = \mathbb Q(\sqrt{D})\) it is shown that the structure of the \(3\)-class groups of the four \(S_3\)-fields \(N_1,\ldots, N_4\) frequently determines the type of principalization of the \(3\)-class group of \(K\) in \(N_1,\ldots, N_4\). This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all \(4 596\) quadratic fields \(K\) with \(3\)-class group of type \((3, 3)\) and discriminant \(-10^6 < D < 10^7\) to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups \(\mathrm{G}_3^2(K)\) on various coclass trees of the coclass graphs \(\mathcal{G}(3, r)\), \(1 \leq r \leq 6\), in the sense of Eick, Leedham-Green, and Newman [B. Eick et al., Int. J. Algebra Comput. 23, No. 5, 1243–1288 (2013; Zbl 1298.20020)].

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R16 Cubic and quartic extensions
11R20 Other abelian and metabelian extensions
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 1298.20020
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Online Encyclopedia of Integer Sequences:

Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 2.
Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (2241).
Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and 3-principalization type (4224).
Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) whose second 3-class group is located on the sporadic part of the coclass graph G(3,2) outside of coclass trees.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.6 (1122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.14 (3122), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type H.4 (2122), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.8 (2234), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type E.9 (2334), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
Minimal absolute discriminants a(n) of complex quadratic fields with 3-class group of type (3,3), 3-principalization type G.16 (2134), second 3-class group G of even nilpotency class cl(G)=2(n+3), and 3-class tower of unknown length at least 3.
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(81,9).
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,95).
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,37).
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to SmallGroup(729,34).
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and second 3-class group isomorphic to either SmallGroup(2187,247)-#1;5 or SmallGroup(2187,247)-#1;9.
Discriminants of real quadratic fields with 3-class tower group <81,10>.
Discriminants of real quadratic fields with 3-class tower group <81,7>
Discriminants of real quadratic fields with second 3-class group <729,49>
Discriminants of real quadratic fields with second 3-class group <729,54>

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