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The second \(p\)-class group of a number field. (English) Zbl 1261.11070

Summary: For a prime \(p\geq 2\) and a number field \(K\) with \(p\)-class group of type \((p,p)\) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=\text{Gal}(F_p^2(K)| K)\) of the second Hilbert \(p\)-class field \(F_p^2(K)\) of \(K\) are determined by the \(p\)-class numbers of the unramified cyclic extensions \(N_{i}| K\), \(1\leq i\leq p + 1\), of relative degree \(p\). In the case of a quadratic field \(K=\mathbb Q(\sqrt D)\) and an odd prime \(p\geq 3\), the invariants of \(G\) are derived from the \(p\)-class numbers of the non-Galois subfields \(L_{i}| \mathbb Q\) of absolute degree \(p\) of the dihedral fields \(N_{i}\). As an application, the structure of the automorphism group \(G=\text{Gal}(F_3^2(K)| K)\) of the second Hilbert 3-class field \(F_3^2(K)\) is analyzed for all quadratic fields \(K\) with discriminant \(-10^6< D < 10^7\) and 3-class group of type \((3, 3)\) by computing their principalization types. The distribution of these metabelian 3-groups \(G\) on the coclass graphs \(\mathcal G(3,r)\), \(1\leq r\leq 6\), in the sense of B. Eick and C. Leedham-Green [Bull. Lond. Math. Soc. 40, No. 2, 274–288 (2008; Zbl 1168.20007)] is investigated.

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11R37 Class field theory
20D15 Finite nilpotent groups, \(p\)-groups
20F12 Commutator calculus
20F14 Derived series, central series, and generalizations for groups

Citations:

Zbl 1168.20007
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Online Encyclopedia of Integer Sequences:

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 E.9 (2334), and second 3-class group G of odd nilpotency class cl(G)=2(n+2)+1.
Fundamental discriminants d uniquely characterizing all complex biquadratic fields Q(sqrt(-3),sqrt(d)) which have 3-class group of type (3,3) and abelian 3-class field tower of length 1.
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 group of type (3,3)
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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