The Primitive Element Theorem (Algebra 3: Lecture 24 Video 3)

Lecture 24: We started this lecture by computing the Galois group of the composite of two Galois extensions as a subgroup of the direct product of the individual Galois groups. We then saw an application focused on the case where the intersection of the two Galois extensions is the base field. We also proved a kind of converse to this statement. We then defined the Galois closure of a finite separable extension and proved that the Galois closure exists. We saw that it is easy to compute the Galois closure of a simple extension. We proved that a finite extension K/F is simple if and only if there are finitely many subfields of K containing F. We used this to prove the Primitive Element Theorem: Any finite separable extension is a simple extension.

Reading: In this lecture we closely followed Section 14.4, starting from page 592.