Hilbert P-class Field Research Articles

On the Invariant Factors of Class Groups in Towers of Number Fields

AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Index-<i>p</i> Abelianization Data of <i>p</i>-Class Tower Groups

Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.

The distribution of second p-class groups on coclass graphs

For a given prime p, Leedham-Green, Newman, and Eick have defined the structure of a directed graph G(p) on the set of all isomorphism classes of finite p-groups. Two vertices are connected by an edge G → H if G is isomorphic to the last lower central quotient H/γc(H) where c = cl(H) denotes the nilpotency class of H. If the condition |H| = p|G| is imposed on the edges, G(p) is partitioned into countably many disjoint subgraphs G(p, r), r ≥ 0, called coclass graphs of p-groups G of coclass r = cc(G) = n− cl(G) where |G| = p. A coclass graph G(p, r) is a forest of finitely many coclass trees T (Gi) with roots Gi, each with a single infinite main line having a pro-p-group of coclass r as its inverse limit, and additionally contains finitely many sporadic groups outside of coclass trees: G(p, r) = (∪i T (Gi)) ∪ G0(p, r). By Artin’s reciprocity law, the second p-class groups Gp(K) = Gal(F 2 p(K)|K) of algebraic number fields K, where Fp(K) denotes the second Hilbert p-class field of K, are vertices of the metabelian skeleton of G(p). Our aim is firstly to provide a general algorithm for determining the structure of Gp(K) for a given number field K by means of number theoretical invariants of the intermediate fields K ≤ N ≤ Fp(K) between K and its first Hilbert p-class field Fp(K) and secondly to show that the arithmetic of special types of base fields K gives rise to selection rules for Gp(K), e.g. • If p = 2 and K is complex quadratic of type (2, 2), there are no selection rules and G(2, 1) is entirely populated by the G2(K), apart from the isolated group C4. • If p = 3 and K is complex quadratic of type (3, 3) or real quadratic of type (3, 3) without total principalization, then either G3(K) is sporadic or lies on an even branch B2k of a coclass tree of an even coclass graph G(3, 2j). • If p ≥ 3, K is quadratic of type (p, p), and Gp(K) is of coclass 1, then K must be real quadratic and Gp(K) lies on an odd branch B2k+1 of the unique coclass tree T (Cp × Cp) of G(p, 1). Our aforementioned new algorithm is based on the family of transfers Vi : G/G ′ → Ui/U ′ i from a metabelian p-group G to all intermediate groups G′ ≤ Ui ≤ G. We prove that the main lines of coclass trees, and all other coclass families arising from the periodicity of branches, share a common transfer kernel type κ(G) = (ker(Vi)) and that κ(G) is determined by the transfer target type τ(G) = (str(Ui/U ′ i)) where str(A) denotes the multiplet of type invariants of an abelian p-group A. Consequently, the structure of Gp(K) and the principalization type of a number field K is determined by the structures of the p-class groups Clp(Ni) of all intermediate fields K ≤ Ni ≤ Fp(K), according to the Artin reciprocity law. We have implemented this algorithm in PARI/GP to determine the structure of the second 3-class groups G3(K) = Gal(F 2 3(K)|K) of the 4 596 quadratic number fields K = Q( √ D) with discriminant −10 < D < 10 and 3-class group Cl3(K) of type (3, 3) and to analyze their distribution on the coclass graphs G(3, r), 1 ≤ r ≤ 6. References. [1] D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. (2011), DOI 10.1007/s00605-010-0277-x. [2] D. C. Mayer, The second p-class group of a number field (preprint 2010). [3] D. C. Mayer, Principalization algorithm via class group structure (preprint 2011).

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

For a prime p ≥ 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 [Formula: see text] of the second Hilbert p-class field [Formula: see text] of K are determined by the p-class numbers of the unramified cyclic extensions Ni|K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field [Formula: see text] and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields Li|ℚ of absolute degree p of the dihedral fields Ni. As an application, the structure of the automorphism group [Formula: see text] of the second Hilbert 3-class field [Formula: see text] is analyzed for all quadratic fields K with discriminant -106&lt; D &lt; 107and 3-class group of type (3, 3) by computing their principalization types. The distribution of these metabelian 3-groups G on the coclass graphs [Formula: see text], 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated.

INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS

AbstractWe use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field $\Q(\exp(2 \pi i/m))$ has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over $\Q(\exp(2 \pi i/m))$ for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of $\Q(\exp(2 \pi i/m))$.

On Cyclic Fields of Odd Prime Degree p with Infinite Hilbert p-Class Field Towers

AbstractLet k be a cyclic extension of odd prime degree p of the field of rational numbers. If t denotes the number of primes that ramify in k, it is known that the Hilbert p-class field tower of k is infinite if t &gt; 3 + 2 . For each t &gt; 2 + , this paper shows that a positive proportion of such fields k have infinite Hilbert p-class field towers.

On embedding problems with restricted ramifications

Let L / k be a Galois extension with Galois group G, and \((\varepsilon ): 1\to A\to E\to G\to 1\) a central extension. We study the existence of the Galois extension M / L / k such that the Galois group Gal(M / k) is isomorphic to E and that M / L is unramified outside S, where S is a finite set of primes of L. As an application, we also study the class number of the Hilbert p-class field.

On Vandiver's conjecture and [formula omitted]p-extensions of [formula omitted](ζ p n)

Suppose K + is the maximal totally real subfield of K = Q( ζ p ) and h + is the class number of K +. Vandiver's conjecture states that p does no divide h +. Each Z p -extension extension K ∞ of K defines a unique Z p -extension R ∞ of R = ▪ x [ p −1] with ▪ k being the ring of integers of K. In this paper we prove for all odd primes p, (1) if p does not divide h +, then each Z p -extension of R has a normal basis over R. Conversely, if each Z p -extension of R has a normal basis over R and if the Iwasawa invariant λ + of the cyclotomic Z p -extension of K + is zero, then p does not divide h +. (2) p does not divide h + if and only if each cyclic extension of K, lying in the minus part H − of the Hilbert p-class field H of K, is contained in a Z p -extension of K.