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Linkage and Essential $p$-Dimension

Adam Chapman

Abstract

We prove that two cyclically linked $p$-algebras of prime degree become inseparably linked under a prime to $p$ extension if and only if the essential $p$-dimension of the pair is 2. We conclude that the essential $p$-dimension of pairs of cyclically linked $p$-algebras is 3 by constructing an example of a pair that does not become inseparably linked under any prime to $p$ extension.

Linkage and Essential $p$-Dimension

Abstract

We prove that two cyclically linked -algebras of prime degree become inseparably linked under a prime to extension if and only if the essential -dimension of the pair is 2. We conclude that the essential -dimension of pairs of cyclically linked -algebras is 3 by constructing an example of a pair that does not become inseparably linked under any prime to extension.
Paper Structure (6 sections, 8 theorems)

This paper contains 6 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Essential Dimension
  4. Kato-Milne Cohomology
  5. Inseparable Linkage and $H_p^3(F)$
  6. Linkage and Essential Dimension

Key Result

Theorem 3.1

The class of $\alpha d\log(\beta)\wedge d\log(\gamma)$ is trivial in $H_p^3(F)$ if and only if $\gamma$ is the norm of an element in the algebra $[\alpha,\beta)_{p,F}$.

Theorems & Definitions (15)

  • Theorem 3.1: Gille:2000
  • Theorem 3.2: Pfister:1995
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • Remark 3.6
  • Theorem 4.1
  • proof
  • ...and 5 more