Field Theory (mathematics) - Closures of A Field

Closures of A Field

Given a field k, various kinds of closures of k may be introduced. For example, the algebraic closure, the separable closure, the cyclic closure et cetera. The idea is always the same: If P is a property of fields, then a P-closure of k is a field K containing k, having property P, and which is minimal in the sense that no proper subfield of K that contains k has property P. For example if we take P(K) to be the property "every nonconstant polynomial f in K has a root in K", then a P-closure of k is just an algebraic closure of k. In general, if P-closures exist for some property P and field k, they are all isomorphic. However, there is in general no preferable isomorphism between two closures.