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A bound on the index of exponent-$4$ algebras in terms of the $u$-invariant

Karim Johannes Becher, Fatma Kader Bingöl

Abstract

For a prime number $p$, an integer $e\geq 2$ and a field $F$ containing a primitive $p^e$-th root of unity, the index of central simple $F$-algebras of exponent $p^e$ is bounded in terms of the $p$-symbol length of $F$. For a nonreal field $F$ of characteristic different from $2$, the index of central simple algebras of exponent $4$ is bounded in terms of the $u$-invariant of $F$. Finally, a new construction for nonreal fields of $u$-invariant $6$ is presented.

A bound on the index of exponent-$4$ algebras in terms of the $u$-invariant

Abstract

For a prime number , an integer and a field containing a primitive -th root of unity, the index of central simple -algebras of exponent is bounded in terms of the -symbol length of . For a nonreal field of characteristic different from , the index of central simple algebras of exponent is bounded in terms of the -invariant of . Finally, a new construction for nonreal fields of -invariant is presented.
Paper Structure (4 sections, 24 theorems, 13 equations)

This paper contains 4 sections, 24 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Multiplication by a power of $p$ in the Brauer group
  3. Multiplying by $2$ in the Brauer group
  4. Examples of fields with $u$-invariant $6$

Key Result

Theorem 1.2

Let $p$ be a prime number and assume that $\operatorname{\mathsf{char}} F=p$. Let $e\in\mathbb{N}^+$. Then $\operatorname{\mathsf{Br}}_{p^e}(F)$ is generated by classes of cyclic $F$-algebras of degree dividing $p^e$.

Theorems & Definitions (52)

  • Theorem 1.2: Albert
  • proof
  • Theorem 1.3: Merkurjev-Suslin
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2: Albert
  • proof
  • Corollary 2.3
  • proof
  • ...and 42 more