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Invariants de Witt des involutions de bas degré en caractéristique 2

Jean-Pierre Tignol

Abstract

A $3$-fold and a $5$-fold quadratic Pfister forms are canonically associated to every symplectic involution on a central simple algebra of degree $8$ over a field of characteristic $2$. The same construction on central simple algebras of degree $4$ associates to every unitary involution a $2$-fold and a $4$-fold Pfister quadratic forms, and to every orthogonal involution a $1$-fold and a $3$-fold quasi-Pfister forms. These forms hold structural information on the algebra with involution.

Invariants de Witt des involutions de bas degré en caractéristique 2

Abstract

A -fold and a -fold quadratic Pfister forms are canonically associated to every symplectic involution on a central simple algebra of degree over a field of characteristic . The same construction on central simple algebras of degree associates to every unitary involution a -fold and a -fold Pfister quadratic forms, and to every orthogonal involution a -fold and a -fold quasi-Pfister forms. These forms hold structural information on the algebra with involution.
Paper Structure (4 sections, 4 theorems, 40 equations)

This paper contains 4 sections, 4 theorems, 40 equations.

Table of Contents

  1. La forme $\operatorname{Srp}_\sigma$
  2. Décomposition orthogonale
  3. Démonstration du théorème \ref{['th:symp']}
  4. Involutions unitaires et orthogonales

Key Result

Proposition 2.1

Pour la forme polaire $b_\sigma$ de $\operatorname{Srp}_\sigma$, l'espace $\operatorname{Symd}(\sigma)$ se décompose en somme orthogonale: $\operatorname{Symd}(\sigma)=L\stackrel{\perp}{\oplus} W_1 \stackrel{\perp}{\oplus} W_2 \stackrel{\perp}{\oplus}W_3$. De plus, pour chaque $i=1$, $2$, $3$ le $L_

Theorems & Definitions (13)

  • proof
  • proof
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • proof
  • Proposition 3.4
  • proof
  • proof
  • ...and 3 more