The Witt group of a dyadic valued field and hereditary orders over discrete valuation rings

This thesis is devoted to the study of different aspects of valuation theory. The first chapter fits into the landscape of the algebraic theory of quadratic forms. A well-known construction consists in forming a group, the Witt group, from the nonsingular quadratic forms over a field. The explicit calculation of this group is possible for some particular fields. In this context, a theorem due to Springer allows to describe the Witt group of a complete field k for a discrete valuation from the Witt group of its residue field, provided that the characteristic of the residue field is different from 2. In the case where the residue characteristic is 2, the obstruction to Springer’s theorem has been studied by various authors. In the first chapter of my thesis, I analyze this obstruction by means of a filtration on the Witt group of k. I also give an interpretation of the successive quotients of the filtration in terms of groups formed from totally singular residue quadratic forms. The second chapter of my thesis contributes to strengthen the connection between valuation theory and noncommutative arithmetic in central simple algebras when the scalar field is equipped with a discrete valuation p. I present the hereditary orders over the ring of integers of p as being the rings of a family of algebra p-norms called gauges. This relation acts as a dictionary allowing to move from the functional point of view, represented by the gauges, to aspects of ring theory. This result extends to central simple involution algebras. Consequently, I also show that the number of hereditary orders the latter possess is influenced by p-anisotropy properties of the involution.(SC - Sciences) -- UCL, 202