Construction and Classification of Some Galois Modules

In our previous paper we describe the Galois module structures of pth-power class groups K/K, where K/F is a cyclic extension of degree p over a field F containing a primitive pth root of unity. Our description relies upon arithmetic invariants associated with K/F . Here we construct field extensions K/F with prescribed arithmetic invariants, thus completing our classification of Galois modules K/K. Let F be a field of characteristic not p containing a primitive pth root of unity ξp. For a cyclic field extension K/F with Galois group Gal(K/F ) of order p, let J = K×/K×p, and let N denote the norm map from K to F . In [MS], we proved that the structure of the Fp[Gal(K/F )]-module J is determined by the following three arithmetic invariants: • d = d(K/F ) := dimFp F×/N(K×), • e = e(K/F ) := dimFp N(K×)/F×p, and • Υ(K/F ) := 1 or 0 according to whether ξp ∈ N(K×) or not. Now if G = Z/pZ, then J may be considered an Fp[G]-module via any isomorphism G ∼= Gal(K/F ), and the module structure of J is independent of the choice of isomorphism. It is a fundamental problem to classify the isomorphism classes of modules J for all K/F in our context. This problem is solved in Theorem 1 below. Corollaries 1 and 2 of Theorem 1 describe all modules J in an explicit way. Date: April 8, 2003. Research supported in part by the Natural Sciences and Engineering Research Council of Canada, and by the special Dean of Science Fund at the University of Western Ontario. Supported by the Mathematical Sciences Research Institute, Berkeley. Research supported in part by National Security Agency grant MDA904-02-10061. 1 2 JÁN MINÁČ AND JOHN SWALLOW In the following theorem we determine the sets of invariants (d, e,Υ) which may be realized by an extension K/F and in so doing classify all Fp[G]-modules J = J(K/F ) up to isomorphism. Theorem 1. Let p be a prime number. For arbitrary cardinal numbers d, e, and for Υ ∈ {0, 1}, there exists a cyclic field extension K/F of degree p containing a primitive pth root of unity with invariants (d, e,Υ) if and only if • if Υ = 0, then 1 ≤ d, • if p > 2 then 1 ≤ e, and • if p = 2 and Υ = 1 then 1 ≤ e. From the theorem above and from [MS, Theorem 3 and Corollary 2] we immediately obtain the following corollaries. We denote by Mi,j the jth cyclic module Fp[G] such that dimFp Mi,j = i, where j is a suitable index. Corollary 1. Let p > 2 be a prime number, and let G be a cyclic group of order p. Then an Fp[G]-module J is realizable as an Fp[Gal(K/F )]module K×/K×p for some cyclic G-extension K/F such that F contains a primitive pth root of unity if and only if there exist cardinal numbers d, e, and Υ ∈ {0, 1} such that (i) If Υ = 0, then 1 ≤ d; (ii) 1 ≤ e; and (iii)