Bases normales d'entiers dans les extensions de Kummer de degré premier

Abstract

If F is a number field, we denote by 𝔇 F its ring of integers. Let E/F be a finite Galois extension of number fields with group G ; a basis of 𝔇 E as 𝔇 F -module of the form a 9 g∈G is called a normal basis of 𝔇 E over 𝔇 F . In this paper we establish an existence criterion for an integral normal basis in a Kummer extension of prime degree, which shows in addition how to construct a normal basis in case it exists. The main tools used in the proof are a formula of Frôhlich for the resolvents and a theorem of Hecke describing the ramification in a Kummer extension of prime degree. As an application, we show how to use our criterion to deduce a normal basis theorem obtained by F. Kawamoto.