Biquadratic Extensions with One Break

Abstract

Abstract. We explicitly describe, in terms of indecomposable Z 2 [G]-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completes work begun in [Elder: Canad.

On the necessity of new ramification breaks

“…The usual ramification invariants are not however sufficient to determine the Galois module structure of ideals. This was observed in [BE02] where we considered biquadratic extensions (the case p = 2) with one break. The work presented here, together with [BE05], stems from our ongoing effort to fully understand the implications of that paper, and to extend its results to arbitrary p. With hindsight we can now say that the insufficiency of the usual ramification filtration is tied to the elementary abelian quotients of consecutive ramification groups G b /G b+1 , but that there is a 'repair.…”

“…Kummer extensions, as we tried to generalize the results of [BE02] from p = 2 to p > 2. In the course of these investigations, truncated exponentiation appeared first within the group ring O T [G], as we worked to prove Lemma 9.…”

On wild ramification in quaternion extensions

“…7s 1 − 2s 2 for i = 1 * , 5s 1 for i = 1, 2s 1 + 3s 2 for i = 2, U i = 8e − 3s 1 for i = 1 * , 1, 8e − 2s 1 − s 2 for i = 2.…”

The ramification group filtrations of certain function field extensions

“…In [6], it is shown that the ramification group filtration of a wildly ramified prime p is uniquely determined by the p-adic valuation of the discriminant of the field extension L/K, when both the field extension degree and the residue characteristic of p are equal to a prime number. When the Galois group is elementary abelian, the Galois module structure of certain ideals is related to the ramification group filtration, see [3,4,5]. Such a relation is investigated when the Galois group is quaternion [7], hence non-abelian.…”