Galois theory: Algebraic closure
Channel: Richard E Borcherds
Date: 2020-12-29

This lecture is part of an online graduate course on Galois theory.

We define the algebraic closure of a field as a sort of splitting field of all polynomials, and check that it is algebraically closed. 

We hen give a topological proof that the field C of complex numbers is algebraically closed, and mention the field of formal Puiseaux series as another example of an algebraically closed field. 

Finally we discuss the uniqueness of algebraic closures of fields, in particular the fact that although any two algebraic closures are isomorphic, the isomorphism is not unique. We mention the analogy with the problem of defining the fundamental group of a topological space.