Characterizing finite fields via minimal ring extensions
David E. Dobbs · Communications in Algebra · 2019
Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that |B|≤N for each ring B such that A⊂B is an inert (minimal ring) extension. It follows that A is a (finite) field ⇔ the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ there exists a minimal ring extension A⊂B such that B is a flat A-module ⇔ there exists a minimal ring extension A⊂B such that B is a local ring and A⊂B is a Galois ring extension.