1. The theorem. Let k be an algebraically closed field. Let A = {a} be an indexing set. Let o=k[xa] be the polynomial ring generated over k by an indexed set of variables xa, aCA. Let a be an ideal of o. A zero of a is a set (ta) of elements Sa in some extension field of k such that f( )=0 for all fCa. (Of course, a polynomial f involves only a finite number of the variables xa.) A zero of a will be called algebraic if all Sa lie in k. The set of all algebraic zeros of an ideal a will be called the variety defined by a. The Hilbert Nullstellensatz is in general not valid if A is an infinite set. We shall prove however the following theorem: