Finite Integral Domains and Finite Fields: Does Size Determine Structure?
2025-05-29
This article explores some well-known results from abstract algebra concerning fields and integral domains. It begins by defining an integral domain and providing examples. The author then proves that every field is an integral domain, every finite integral domain is a field, but infinite integral domains may or may not be fields. Two proofs are given for the finite case, highlighting the fascinating interplay between finiteness and algebraic structure.