Normal Bases in Galois Extensions of Number Fields

Abstract

The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers OK of a number field was a free Z-module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F. A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.

Galois structure of Zariski cohomology for weakly ramified covers of curves

“…Unfortunately, he does not state the general case of the if-part of Theorem (1.1) which is essential for this paper. Though it certainly can be proved with the methods he has developed in his papers [Ul1], [Ul2] and [Ul3], we here give a coherent and self-contained proof of Theorem (1.1) for the reader's convenience. (b) In the geometric case, Pink has given a "global proof" for the fact that L/K is weakly ramified, if and only if m L is O K [G]-free (see Corollary 3.6 in [Pi]); to be precise, the if-direction is proved there only under the additional assumption G = G 1 .…”

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

“…By the above one to one correspondence between ideals with a normal basis and orders with a normal basis and by Jakubec, Kostra [1], Ullom [3] the following two theorems hold. …”

An infinite family of elliptic curves and Galois module structure

“…p 2 O L ) is a cohomologically trivial Gal(L/F )-module (see [Ul,Corollary 1.4]). Finally the proof follows by computing the ideal factorization of θ using Proposition 2.2 for case (7).…”