Eva Bayer--Fluckiger: Isometries of quadratic spaces over global fields

eva.bayer@epfl.ch

Submission: 2011, Dec 4

Let $k$ be a field of characteristic not 2. A quadratic space is a non--degenerate symmetric bilinear form $q: V \times V \to k$ defined on a finite dimensional $k$--vector space $V$, and an isometry of $(V,q)$ is an element of $SO(q)$, in other words an isomorphism $t : V \to V$ such that $q(tx,ty) = q(x,y)$ for all $x,y \in V$ and that $det (t) = 1$. Milnor investigated isometries of quadratic spaces, and raised the following question :

Question. Let $q$ be a quadratic space over $k$, and let $f \in k[X]$ be an irreducible polynomial. How can we tell whether $q$ has an isometry with minimal polynomial $f$ ?

The purpose of the present paper is to study and give an answer to this question for global fields.

2010 Mathematics Subject Classification: 11E12

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