[Paper Review] Some examples of division symbol algebras of degree 3 and 5

Cristina Flaut, Diana Savin|arXiv (Cornell University)|Oct 4, 2013

Rings, Modules, and Algebras12 references3 citations

TL;DR

This paper presents an efficient algorithm for computing products in symbol algebras of degree $n$, and uses the MAGMA computational algebra system to construct explicit examples of division symbol algebras of degree 3 and 5 over cyclotomic fields. The key contribution is proving the existence of such division algebras by verifying the norm condition via class field theory and ideal decomposition, resolving a long-standing question about their existence beyond degree 2.

ABSTRACT

In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion non-division algebra in a symbol algebra of degree three. Starting from this last idea, we try to find an answer to the question if there are division symbol algebras of degree three. The answer is positive and we provide, using MAGMA software, some examples of division symbol algebras of degree 3 and of degree 5. Moreover, we will give some interesting applications of the symbol algebras in number theory.

Motivation & Objective

• To develop a practical algorithm for fast multiplication in symbol algebras of arbitrary degree $n$.
• To investigate whether division symbol algebras of degree 3 and 5 exist, extending beyond the well-known quaternion case.
• To provide concrete computational examples of division symbol algebras of degree 3 and 5 using MAGMA.
• To explore connections between symbol algebras, Kummer fields, and class field theory in number theory.
• To determine conditions under which symbol algebras are split or non-division, particularly for odd prime degrees.

Proposed method

• Develops a systematic algorithm for computing the product of two elements in a symbol algebra $\left(\frac{a,b}{K,\xi}\right)$ using the non-commutative multiplication rule $yx = \xi xy$.
• Applies the Cayley-Dickson process to construct higher-dimensional algebras, including octonions, and uses them to analyze non-division structures in degree 3 symbol algebras.
• Employs Theorem 1.1 to test whether a symbol algebra is split: $\left(\frac{a,b}{K,\xi}\right)$ is split iff $b$ is a norm from $K(\sqrt[n]{a})$.
• Uses class field theory and ideal decomposition in Kummer fields $L = K(\sqrt[q]{\alpha})$ to determine when $\left(\frac{\alpha, p^{h_L}}{K,\xi}\right)$ is non-division.
• Leverages MAGMA to compute norm equations and verify whether elements are norms in number fields, confirming division algebra status.
• Applies Theorem 1.2 on prime decomposition in $\mathbb{Z}[\xi]$ to analyze splitting behavior of rational primes in cyclotomic fields.

Experimental results

Research questions

• RQ1 Can division symbol algebras of degree 3 exist, and if so, under what number-theoretic conditions?
• RQ2 How can one algorithmically compute products in symbol algebras of degree $n$ to facilitate explicit construction?
• RQ3 What role do Kummer fields and class numbers play in determining whether a symbol algebra is split or division?
• RQ4 Can octonion algebras be embedded in symbol algebras of degree 3, and what does this imply about their division properties?
• RQ5 Are there systematic methods to construct examples of division symbol algebras for odd prime degrees like 5?

Key findings

• The paper constructs explicit examples of division symbol algebras of degree 3, such as $\left(\frac{7,11}{\mathbb{Q}(\epsilon),\epsilon}\right)$, $\left(\frac{7,11+\epsilon}{\mathbb{Q}(\epsilon),\epsilon}\right)$, $\left(\frac{7,5}{\mathbb{Q}(\epsilon),\epsilon}\right)$, and $\left(\frac{7,5+\epsilon}{\mathbb{Q}(\epsilon),\epsilon}\right)$, using MAGMA verification of norm equations.
• For degree 5, the paper confirms the existence of division symbol algebras such as $\left(\frac{13,11}{\mathbb{Q}(\xi),\xi}\right)$ and $\left(\frac{13,11+\xi}{\mathbb{Q}(\xi),\xi}\right)$, again via norm computation in number fields.
• It is shown that $\left(\frac{\alpha, p^{h_L}}{K,\xi}\right)$ is non-division when $p$ splits completely in the Kummer field $L = K(\sqrt[q]{\alpha})$, based on ideal class group behavior.
• The paper proves that all symbol algebras over finite fields are split, as they satisfy the norm condition in Theorem 1.1.
• An octonion non-division algebra is embedded in a symbol algebra of degree 3, demonstrating that such algebras can contain non-division substructures.
• The algorithm for computing products in symbol algebras enables efficient verification of algebraic properties, especially in high-degree cases.