Definable equivalence relations on algebraically closed fields

Abstract

This article was inspired by the question: is there a definable equivalence relation on the field of complex numbers, each of whose equivalence classes has exactly two elements? The answer turned out to be no , as we now explain in greater detail. Let Κ be an algebraically closed field and let E be a definable equivalence relation on Κ. [Note: By “definable” we will always mean “definable with parameters”.] Either E has one cofinite class, or all classes are finite and there is a number d such that all but a finite set of classes have cardinality d . In the latter case let B be the finite set of elements of Κ which are not in a class of size d . We prove the following result. Theorem 1. a) If char( Κ ) = 0 or char( Κ ) = p > d , then ∣ B ∣ ≡ 1 (mod d ). b) If char( Κ ) = 2 and d = 2, then ∣ B ∣ ≡ 0 (mod 2). c) If char( Κ ) = p > 2 and d = p + s , where 1 ≤ s ≤ p /2, then ∣ B ∣ ≡ p + 1 (mod d ). Furthermore , a)−c) are the only restrictions on ≡ B ≡. If one is in the right mood, one can view this theorem as saying that the “algebraic cardinality” of the complex numbers is congruent to 1 (mod n ) for every n . §1 contains a reduction of the problem to the special case where E is induced by a rational function in one variable. §2 contains the main calculations and the proofs of a)−c). §3 contains eight families of examples showing that all else is possible. In §4 we prove an analogous result for real closed fields.