The previous lecture in this series is here. The next lecture in this series is here.
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 then 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.
This video is part of a lecture course by Richard Borcherds from 2020-21.