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Symbol length in positive characteristic

Fatma Kader Bingöl

Abstract

We show that any central simple algebra of exponent $p$ in prime characteristic $p$ that is split by a $p$-extension of degree $p^n$ is Brauer equivalent to a tensor product of $2\cdot p^{n-1}-1$ cyclic algebras of degree $p$. If $p=2$ and $n\geq3$, we improve this result by showing that such an algebra is Brauer equivalent to a tensor product of $5\cdot2^{n-3}-1$ quaternion algebras. Furthermore, we provide new proofs for some bounds on the minimum number of cyclic algebras of degree $p$ that is needed to represent Brauer classes of central simple algebras of exponent $p$ in prime characteristic $p$, which have previously been obtained by different methods.

Symbol length in positive characteristic

Abstract

We show that any central simple algebra of exponent in prime characteristic that is split by a -extension of degree is Brauer equivalent to a tensor product of cyclic algebras of degree . If and , we improve this result by showing that such an algebra is Brauer equivalent to a tensor product of quaternion algebras. Furthermore, we provide new proofs for some bounds on the minimum number of cyclic algebras of degree that is needed to represent Brauer classes of central simple algebras of exponent in prime characteristic , which have previously been obtained by different methods.
Paper Structure (6 sections, 24 theorems, 11 equations)

This paper contains 6 sections, 24 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Some classical results
  3. The frobenius morphism
  4. Cyclic algebras of exponent $p$ in characteristic $p$
  5. Algebras of exponent $p$ in characteristic $p$
  6. Splitting by $p$-extensions

Key Result

Theorem 1.1

Let $n\in\mathbb{N}_+$. Assume $F$ contains a primitive $n$-th root of unity or $n$ is a power of $\operatorname{\mathsf{char}} F$. Then $\operatorname{\mathsf{Br}}_{n}(F)$ is generated by the classes of cyclic algebras of degree $n$.

Theorems & Definitions (46)

  • Theorem 1.1: A. A. Albert, A. S. Merkurjev, A. A. Suslin Alb39, MS1982
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: M. Florence Flo13
  • Theorem 1.5: Mammone--Merkurjev
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Theorem 2.2: Albert
  • proof
  • ...and 36 more