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Splitting quaternion algebras defined over a finite field extension

Karim Johannes Becher, Fatma Kader Bingöl, David B. Leep

Abstract

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every central simple algebra of degree 16 is split by a 2-extension of degree at most 2^{16}.

Splitting quaternion algebras defined over a finite field extension

Abstract

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every central simple algebra of degree 16 is split by a 2-extension of degree at most 2^{16}.

Paper Structure

This paper contains 5 sections, 17 theorems, 17 equations.

Table of Contents

  1. introduction
  2. System of quadratic forms
  3. Quaternion algebras defined over a finite extension
  4. The corestriction of a quaternion algebra
  5. Splitting fields for algebras of exponent $2$

Key Result

Theorem 2.1

Let $r\geqslant 1$ and let $\varphi$ be a system of $r$ quadratic forms defined on an $F$-vector space of dimension at least $\frac{r(r+1)}{2}+1$. There exists a generalized $2$-extension $F'/F$ with $[F':F]\leqslant2^r$ such that $\varphi_{F'}$ is isotropic.

Theorems & Definitions (34)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 24 more