From Finite Integral Domains to Finite Fields

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 of Integral Domain 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. Equivalently, an integral domain is a commutative ring, with distinct additive and multiplicative identities, such that if the product of two elements is zero, then one of the elements must be zero. Using standard notation, we can write that a commutative ring \( R \) is an integral do ... Read full article.