[Paper Review] Determination of division algebras with 243 elements

I. F. Rúa, Elías F. Combarro+1 more|arXiv (Cornell University)|Oct 1, 2010

Coding theory and cryptography13 references3 citations

TL;DR

This paper presents a complete computer-assisted classification of finite nonassociative division algebras (semifields) with 243 elements, up to isotopy, using matrix-based representations and algorithms from prior work. It identifies nine non-isotopic semifields, including two new ones not previously known, and fully characterizes their algebraic invariants such as center, nuclei, automorphism groups, and orbit structures in the fundamental triangle of isotopes.

ABSTRACT

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.

Motivation & Objective

• To complete the classification of finite semifields of order 243, a remaining case in the program to classify semifield planes of order 256 or less.
• To extend computational methods for classifying finite semifields beyond orders 16, 32, 64, and 81.
• To identify and characterize all isotopy classes of semifields with 243 elements using a matrix-based representation and algorithmic search.
• To compute key invariants such as center, nuclei, autotopism group orders, and orbit structures in the fundamental triangle for each class.
• To verify and extend known constructions of semifields, including Albert’s twisted fields and Coulter-Matthews type algebras, and discover new ones.

Proposed method

• Represent each semifield via a standard basis of 5 invertible 5×5 matrices over 𝔽₃, satisfying specific conditions: A₁ is the identity, all nontrivial linear combinations are invertible, and Aᵢ has a 1 in the i-th diagonal position.
• Encode each matrix as a base-3 integer using its entries in a specific order to enable efficient computation and storage.
• Use the algorithmic framework from prior work on semifields of order 64 to systematically search for valid 5-tuples of matrices satisfying the semifield axioms.
• Apply isotopy reduction by fixing A₂ to one of six canonical forms (companion matrices of degree-5 or block-diagonal polynomials) to avoid redundant enumeration.
• Compute invariants such as center, nuclei, isomorphism group order, and orbit structure of the fundamental triangle (Lₓ, L∞, Ly) under S₃ action.
• Use computational tools and supercomputing resources to verify and enumerate all isotopy classes, identifying two new semifields (classes VIII and IX).

Experimental results

Research questions

• RQ1 How many non-isotopic semifields of order 243 exist, and what are their structural invariants?
• RQ2 Which of the semifields of order 243 are new, and how do they differ from known constructions like Albert’s twisted fields or Coulter-Matthews algebras?
• RQ3 What are the orders and structures of the automorphism and autotopism groups for each semifield class?
• RQ4 How do the orbit structures of the fundamental triangle (Lₓ, L∞, Ly) vary across the isotopy classes?
• RQ5 Can the classification of semifields of order 243 be completed using a matrix-based computational approach with efficient encoding and search?

Key findings

• There are exactly nine isotopy classes of finite semifields with 243 elements.
• Two of these classes (VIII and IX) are new and not isomorphic to any previously known semifields.
• The nine classes include the commutative field 𝔽₂₄₃, Albert’s twisted fields (classes II and III), and other known families such as Coulter-Matthews and Ding-Yuan.
• The autotopism group orders range from 4 (classes VIII and IX) to 292,820 (class I), with class I being the field and having the largest symmetry.
• The fundamental triangle orbit structure varies: classes I–VII have orbits of the form 2[1]+1[242] or 2[1]+1[2]+24[10], while classes VIII and IX have 2[1]+121[2], indicating higher symmetry in their isotopy orbits.
• All semifields have center and nucleus of size 3, except class I, which has full nucleus and center of size 243, confirming it is the field.