• Frohlich's conjecture
• Hom Description
• quaternion
• tamely ramified
• Theory of Numbers

In this thesis I present some fundamental results concerning the study of the normal integral basis problem, in the case of tamely ramified quaternion extensions over the rational field. These results, principally due to A. Fröhlich, are particularly relevant since they represent the starting point of deep and substantial further discoveries on the subject.
In 1972 Fröhlich introduced and subsequently settled a conjecture, establishing a surprising connection between the existence of a normal integral basis and the sign of the Artin root number of the unique order 2 irreducible character, for tame quaternion extensions over the field Q.
The present work is based on a series of lectures held by Prof. Ph. Cassou-Noguès at the University of Pisa in the year 2014. The aim of those lectures was to give a rapid insight into the techniques developed so far to study Fröhlich's conjecture and similar problems.
I propose here two different proofs of Fröhlich's conjecture. The first one runs through the results found by J.Martinet in Modules sur l'algébre du groupe quaternionien (1971) and by A.Fröhlich in Artin root numbers and normal integral bases for quaternion fields (1972), and is essentially a "hand-made" computation. The second strategy of proof is far more instructive and linear, although it requires a huge theoretical background (see Fröhlich: Arithmetic and Galois module structure for tame extensions, 1976). It makes use of some innovative and brilliant theory developed by Fröhlich and others, which allows an explicit description of the class group of any order, via character theory, the so called Hom description, which I introduce apart in Chapter 5.