The theory of integrally closed domains is not finitely axiomatizable

It is well-known that the theory of algebraically closed fields is not finitely axiomatizable. In this note, we prove that the theory of integrally closed integral domains is also not finitely axiomatizable. All rings in this paper are assumed commutative with identity. Let L be the language of rings, that is, the language whose signature consists of equality, binary function symbols + and ·, and constants 0 and 1. The theory TAC of algebraically closed fields is the set of consequences of the collection ∑ of sentences comprised of (1) the conjunction βF of the field axioms, and (2) for every positive integer n, a sentence βn asserting that every polynomial of degree n has a root. It is well-known that any two algebraically closed fields of the same uncountable cardinality and characteristic are isomorphic (this follows immediately from a famous theorem of Steinitz). Therefore, by the Los-Vaught Test (as is also well-known), the theory of algebraically closed fields of characteristic p is complete, where p is either 0 or a prime. It is also known that TAC is not finitely axiomatizable ([1], Theorem 3.22). We present a simplified proof of Theorem 3.22 below. Proposition 1 ([1], Theorems 3.21 and 3.22). The theory of algebraically closed fields is not finitely axiomatizable. Sketch of proof. By the Compactness Theorem, it suffices to prove that every finite subset of ∑ has a model which is not an algebraically closed field. Toward this end, we need only show that for every positive integer k, there exists a field F which is not algebraically closed, but for which every polynomial of positive degree d ≤ k has a root in F . Clearly we may assume k > 1. Let S be the multiplicative semigroup generated by the collection of all primes p ≤ k. Now fix an aribtrary prime q, and 2010 Mathematics Subject Classification: 13A99, 03C20 (primary); 12E20 (secondary).