Hopf-Galois module structure of tame biquadratic extensions

Abstract

In [14] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields L/K with group G≅C 2 ×C 2 and study in detail the local and global structure of the ring of integers 𝔒 L as a module over its associated order 𝔄 H in each of the Hopf algebras H giving a nonclassical Hopf-Galois structure on the extension. The results of [14] imply that 𝔒 L is locally free over each 𝔄 H , and we derive necessary and sufficient conditions for 𝔒 L to be free over each 𝔄 H . In particular, we consider the case K=ℚ, and construct extensions exhibiting a variety of global behaviour, which implies that the direct analogue of the Hilbert-Speiser theorem does not hold.