Module structure of integers in metacyclic extensions
by
James E. Carter
College of Charleston

Let L/k be a finite extension of algebraic number fields. Let o_L and o denote the rings of integers in L and k, respectively. As an o-module o_L is completely determined by [L:k] and its Steinitz class C(L, k). Now let G be a finite group containing a normal subgroup H. Then we have an exact sequence of groups

\Sigma: 1 --> H --> G --> G/H --> 1.

With k as above, fix a normal extension E/k with Galois group Gal(E/k) =~ G/H. Suppose L/k is a normal extension such that E subset or equal L, and there exists an isomorphism \phi_L:Gal(L/k) --> G. Furthermore, assume E is the subfield of L fixed by \phi_L^-1(H). An extension L/k as just described will be called a G-extension with respect to E/k and \Sigma. As L varies over all such extensions of k, C(L, k) varies over a subset R(E/k, \Sigma) of the class group C(k) of k. If we consider only tamely ramified extensions then we denote this set by R_t(E/k, \Sigma).

In the present paper we consider the following special case. Let p and q be distinct odd prime numbers and assume k contains the multiplicative group of pq-th roots of unity. Let G be the metacyclic group of order pq given in terms of generators and relations by

<\sigma, \tau| \sigmap=1, \tauq=1, \tau\sigma\tau-1=\sigmar >

where r is a primitive q-th root of unity mod p (and hence, p \equiv 1 mod q). Let s be the unique integer in {2, 3, ... , p-1} such that sr \equiv 1 mod p. Then s is also a primitive q-th root of unity mod p. Hence, s^q=1+tp for some positive integer t. The cyclic subgroup <\sigma> of G generated by \sigma is a normal subgroup of G and we have an exact sequence of groups

\Sigma: 1 --> <\sigma> --> G --> G/<\sigma> --> 1.

Fix, once and for all, a tamely ramified normal extension E/k with Gal(E/k) =~ G/<\sigma>. Furthermore, assume p and q are such that t\not \equiv 0 mod p. Under this condition we will determine R_t(E/k, \Sigma). In particular, we show that if o_E is free as an o-module, then R_t(E/k, \Sigma) is a subgroup of C(k).

Date received: April 5, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caar-02.