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 O K 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.