Separable and Galois extensions

Symmetry is most useful when you have a highly symmetric object. In the theory of field extensions, this happens when you have Galois extensions, the topic of this video. We go through two obstructions to having a sufficiently large Galois group, which correspond to the problems of not enough roots ( not being normal) and repeated roots (not being separable). We study the condition of separability showing that it always holds in characteristic zero by using the classic argument invoking derivatives of polynomials over arbitrary fields. Finally, we show how finite Galois extensions correspond to separable normal extensions, and show how, given a separable extension, we can enlarge it to a Galois extension.