[Paper Review] When is the multiplicative group of a field indecomposable?

Sunil K. Chebolu, Keir Lockridge|arXiv (Cornell University)|Jul 13, 2014

Finite Group Theory Research10 references3 citations

TL;DR

This paper classifies finite fields and finite commutative rings whose multiplicative groups are indecomposable—meaning they cannot be expressed as a nontrivial direct product of subgroups. Using elementary group theory and Catalan’s Conjecture, it shows that finite fields with indecomposable multiplicative groups are those of order $\mathbb{F}_q$ where $q-1$ is a prime power, including Fermat and Mersenne primes. For infinite fields of characteristic $\neq 2$, the multiplicative group is always decomposable, and a complete classification of finite commutative rings with indecomposable unit groups is provided.

ABSTRACT

The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We conjecture that this is also true for infinite fields of characteristic 2 and we narrow the class of possible counter-examples. Finally, using the classification of finite commutative primary rings with cyclic multiplicative groups, we determine all finite commutative rings with indecomposable multiplicative groups.

Motivation & Objective

• To determine which finite fields have indecomposable multiplicative groups.
• To extend the classification to infinite fields, particularly those of characteristic not equal to 2.
• To classify all finite commutative rings with indecomposable multiplicative groups of units.
• To leverage known classifications of rings with cyclic unit groups to derive the full list of rings with indecomposable unit groups.

Proposed method

• Apply the structure theorem for finite abelian groups to analyze when a cyclic group of order $p^r - 1$ is indecomposable.
• Use elementary number theory to solve $p^r - 1 = q^n$ for prime powers, distinguishing cases for Fermat and Mersenne primes.
• Apply Catalan’s Conjecture (proved by Mihăilescu) as a corollary to simplify the classification of solutions to $p^r - 1 = q^n$.
• Use the decomposition of finite commutative rings into products of local Artinian rings to reduce the problem to primary rings.
• Leverage the classification of finite commutative primary rings with cyclic unit groups from Pearson and Schneider (1970).
• Verify indecomposability by checking the absence of nontrivial idempotents in candidate rings and confirming unit group structure.

Experimental results

Research questions

• RQ1 For which finite fields is the multiplicative group indecomposable?
• RQ2 Can the classification of finite fields with indecomposable multiplicative groups be derived from Catalan’s Conjecture?
• RQ3 Is the multiplicative group of every infinite field of characteristic not equal to 2 decomposable?
• RQ4 Which finite commutative rings have indecomposable multiplicative groups of units?
• RQ5 What is the complete list of finite commutative indecomposable rings with indecomposable unit groups?

Key findings

• The multiplicative group of a finite field $\mathbb{F}_q$ is indecomposable if and only if $q-1$ is a prime power, including 1.
• Finite fields with indecomposable multiplicative groups are $\mathbb{F}_2$, $\mathbb{F}_9$, $\mathbb{F}_p$ for Fermat primes $p$, and $\mathbb{F}_q$ where $q-1$ is a Mersenne prime.
• Any infinite field of characteristic not equal to 2 has a decomposable multiplicative group.
• The only finite commutative rings with indecomposable multiplicative groups are products of $\mathbb{Z}_2$ and exactly one of eight specific indecomposable rings.
• The eight indecomposable rings are $\mathbb{F}_2$, $\mathbb{F}_9$, $\mathbb{F}_p$ for Fermat primes $p$, $\mathbb{F}_q$ for Mersenne primes $q$, $\mathbb{Z}_4$, $\mathbb{F}_2[x]/(x^2)$, $\mathbb{F}_2[x]/(x^3)$, and $\mathbb{Z}_4[x]/(2x, x^2-2)$.
• Each of these eight rings is indecomposable as a ring and has a unit group of prime power order or trivial order.