can be generalized using transfinite induction: This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures. Generalizations Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case. Relative algebraic closures Given a field k and a field K containing k, one defines the relative algebraic closure of k in K to be the subfield of K consisting of all elements of K that are algebraic over k, that is all elements of K that are a root of some nonzero polynomial with coefficients in k. See also Integral element Lüroth's theorem Galois extension Separable extension Normal extension Notes References Field extensions