From Finite Integral Domains to Finite Fields

Summary

From Finite Integral Domains to Finite Fields By Susam Pal on 25 May 2025 In this article, we explore a few well-known results from abstract algebra pertaining to fields and integral domains. We ask ourselves whether every field is an integral domain, and whether every integral domain is a field. We begin with the definition of an integral domain, discuss a few established results, and then proceed to answer these questions. Familiarity with algebraic structures such as rings and fields is assumed. Contents Definition and Examples An integral domain is a commutative ring, with distinct additive and multiplicative identities, in which the product of any two non-zero elements is also non-zero. For example, the ring of integers \( \mathbb{Z} \) is an integral domain since the product of two non-zero integers is non-zero. The field of rational numbers \( \mathbb{Q} \) is also an integral domain. The ring of polynomials in the indeterminate \( t \) with coefficients in an integral domain \( R, \) denoted \( R[t], \) is an integral domain as well. However, the ring of integers modulo 6, denoted \( \mathbb{Z}_6, \) is not an integral domain since \( 2 \cdot 3 = 0 \) in \( \mathbb{Z}_6. \) Known Results For the sake of brevity, we assume the following known results. Proposition 1. Let \( R \) be a ring. Then, for all \( a \in R, \) we have \[ a \cdot 0 = 0 \cdot a = 0. \] Proposition 2. Let \( D \) be an integral domain. Then, for all \( a, b, c \in D \) such that \( a \ne 0, \) we have \[ a \cdot b = a \cdot c \implies b = c. \] The second result is also known as the cancellation property of integral domains. Every Field Is an Integral Domain We now show that every field is indeed an integral domain. Let \( F \) be a field, and let \( a, b \in F \) such that \( ab = 0. \) There are two cases to consider: \( a = 0 \) and \( a \ne 0. \) If \( a = 0, \) then indeed \( ab = 0 \) by Proposition 1. Now suppose \( a \ne 0. \) Then by the properties of fields, there exists a multi...