Abstract |
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[eng] The thesis is divided into two parts reflecting various aspects of valuation theory:
1) Construction of gauges on Clifford algebras.
Let (V,q) be a quadratic space over a field F of arbitrary characteristic. Suppose that V is a normed vector space. We construct a canonical induced norm on the Clifford algebra C(V,q), and give conditions for this norm to be a gauge.
To do this, we give a general construction of a value function on a quotient of vector spaces and a criterion for splitting direct sums. Secondly, we develop the notion of value function bounded or compatible with a bilinear (resp. quadratic) form. For this, we introduce a new kind of radical for quadratic spaces with value function. Using the results above we build a canonical surmultiplicative norm on the Clifford algebra of a quadratic form bounded by a norm. Moreover, if the norm is tame with respect to the quadratic form, then the norm on the Clifford algebra is a gauge.
We apply these results to prove that if the base field is complete for a discrete valuation, then a quadratic form of type E_7 cannot be both a transfer of a totally ramified extension and a transfer of an inertial extension.
2) Construction of noncrossed product division algebras with Mal’cev Neumann rings.
We develop a new way to construct noncrossed product division algebras with Mal’cev Neumann series ring. For this, we establish a condition for a Mal’cev-Neumann ring to be a division algebra, by proving that there exists a valuation on this ring. Moreover, we give two detailed constructions of noncrossed product division algebras, one of degree p^2 for p an odd prime and one of degree 8. These constructions are based on constructions of Hanke but we avoid delicate arguments involving outer automorphisms by using local invariants of division algebras over global fields. |