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Bounds on the hermitian u-invariants under quadratic field extensions

Karim Johannes Becher, Fatma Kader Bingöl

Abstract

The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants under a quadratic field extension. This is applied to obtain bounds in terms of the index and the u-invariant of the base field.

Bounds on the hermitian u-invariants under quadratic field extensions

Abstract

The hermitian u-invariants of a central simple algebra with involution are studied. In this context, a new technique is obtained to give bounds for the behavior of these invariants under a quadratic field extension. This is applied to obtain bounds in terms of the index and the u-invariant of the base field.
Paper Structure (6 sections, 26 theorems, 30 equations)

This paper contains 6 sections, 26 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Hermitian forms and involutions
  3. Hermitian $u$-invariants
  4. Isotropy via quadratic reduction
  5. Bounds on the unitary $u$-invariant
  6. Bounds on the orthogonal $u$-invariant

Key Result

Theorem 2.1

Let $A$ be a central simple $F$-algebra. There exists an orthogonal involution on $A$ if and only if $\operatorname{\mathsf{exp}} A\leqslant 2$. There exists a symplectic involution on $A$ if and only if $\operatorname{\mathsf{exp}} A\leqslant 2$ and $\operatorname{\mathsf{deg}} A$ is even.

Theorems & Definitions (57)

  • Theorem 2.1: Albert
  • proof
  • Theorem 2.2: Albert-Riehm-Scharlau
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 47 more