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A classification of the division algebras that are isotopic to a cyclic Galois field extension

Susanne Pumpluen

TL;DR

This work classifies n-dimensional division algebras that are principal Albert isotopes of cyclic field extensions, with a particularly sharp, cubic-case classification. It develops a framework around the unital heart K and Kaplansky's trick to reduce the problem to isotopy questions between K^{(f,g)} algebras, using invertible linear maps and crossed products to model End_F(V) and the associated norms. For cubic hearts, the authors provide a complete, nonoverlapping enumeration of isomorphism types and establish concrete criteria to decide when two such algebras are isomorphic, achieving a tight, distinguishable classification. The results extend to general cyclic Galois hearts, outlining an enumeration scheme that depends on the structure of f' and invariant subsets, and clarifying how isomorphism classes interrelate under Aut(K/F) and Gal(K/F) actions. Altogether, the paper contributes a principled, structurally transparent approach to nonassociative division algebras tied to cyclic Galois extensions with broad implications for principal Albert isotopy classifications in higher dimensions.

Abstract

We classify all division algebras that are principal Albert isotopes of a cyclic Galois field extension of degree $n>2$ up to isomorphisms. We achieve a ``tight'' classification when the cyclic Galois field extension is cubic. The classification is ``tight'' in the sense that the list of algebras has features that make it easy to distinguish non-isomorphic ones.

A classification of the division algebras that are isotopic to a cyclic Galois field extension

TL;DR

This work classifies n-dimensional division algebras that are principal Albert isotopes of cyclic field extensions, with a particularly sharp, cubic-case classification. It develops a framework around the unital heart K and Kaplansky's trick to reduce the problem to isotopy questions between K^{(f,g)} algebras, using invertible linear maps and crossed products to model End_F(V) and the associated norms. For cubic hearts, the authors provide a complete, nonoverlapping enumeration of isomorphism types and establish concrete criteria to decide when two such algebras are isomorphic, achieving a tight, distinguishable classification. The results extend to general cyclic Galois hearts, outlining an enumeration scheme that depends on the structure of f' and invariant subsets, and clarifying how isomorphism classes interrelate under Aut(K/F) and Gal(K/F) actions. Altogether, the paper contributes a principled, structurally transparent approach to nonassociative division algebras tied to cyclic Galois extensions with broad implications for principal Albert isotopy classifications in higher dimensions.

Abstract

We classify all division algebras that are principal Albert isotopes of a cyclic Galois field extension of degree up to isomorphisms. We achieve a ``tight'' classification when the cyclic Galois field extension is cubic. The classification is ``tight'' in the sense that the list of algebras has features that make it easy to distinguish non-isomorphic ones.
Paper Structure (8 sections, 25 theorems, 76 equations)

This paper contains 8 sections, 25 theorems, 76 equations.

Table of Contents

  1. Singular and regular algebras and the unital heart
  2. Isotopes and Kaplansky's trick
  3. Invertible linear maps
  4. Invertible linear maps for Galois $C_n$-algebras
  5. Invertible linear maps for any abelian Galois field extension
  6. Regular algebras with a Galois $C_n$-algebra as unital heart
  7. A classification of division algebras with a cubic Galois field extension as the unital heart
  8. Towards a classifications of algebras whose unital heart is a cyclic Galois field extension

Key Result

Lemma 1.1

For $A,B\in {\rm Alg}(M)$, $(f,g)\in S_0(M)$, every isomorphism $\varphi:A\rightarrow B$ is also an isomorphism $\varphi: A^{(f,g)}\rightarrow B^{(f',g')}$ where $f'=\varphi f\varphi^{-1}$, and $g'=\varphi g\varphi^{-1}$.

Theorems & Definitions (42)

  • Lemma 1.1
  • Proposition 1.2
  • Lemma 1.3
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • proof
  • Corollary 1.8
  • ...and 32 more