A minimal ring extension of a large finite local prime ring is probably ramified.

Given any minimal ring extension k ⊂ L of finite fields, several families of examples are constructed of a finite local (commutative unital) ring A which is not a field, with a (necessarily finite) inert (minimal ring) extension A ⊂ B (so that B is a separable A -algebra), such that A ⊂ B is not a Galois extension and the residue field of A (respectively, B) is k (respectively, L). These results refute an assertion of G. Ganske and McDonald stating that if R ⊆ S are finite local rings such that S is a separable R -algebra, then R ⊆ S is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let (A , M) be a finite special principal ideal ring (SPIR), but not a field, such that M has index of nilpotency α (≥ 2). Impose the uniform distribution on the (finite) set of (A -algebra) isomorphism classes of the minimal ring extensions of A. If 2 ∈ M (for instance, if A ≅ ℤ / 2 α ℤ), the probability that a random isomorphism class consists of ramified extensions of A is at least 2 / 3 ; if 2 ∉ M (for instance, if A ≅ ℤ / p α ℤ for some odd prime p), the corresponding probability is at least 3 / 4. Additional applications, examples and historical remarks are given. [ABSTRACT FROM AUTHOR]