The Definition of a Normal Extension (Algebra 3: Lecture 19 Video 3)

Lecture 19: We started this lecture by stating, but not proving, a main result about a finite subgroup G of Aut(K) and [K:F], where F is the fixed field of G.  We used this result to prove several corollaries, for example, K/F is Galois if and only if the fixed field of Aut(K/F) is F.  We proved that if K/F is Galois, then every irreducible in F[x] that has a root in K is separable and has all of its roots in K.  We discussed the definition of a normal extension in Section 13.4 and explained that many other references give a different definition.  We explained, but did not prove, that for finite extensions these notions of normal coincide.  We defined the Galois conjugates of an element and saw that in a Galois extension K/F, the Galois group acts transitively on the set of roots of the minimal polynomial of an element alpha.  We proved that K/F is Galois if and only if K is the splitting field over F of some separable polynomial in F[x].  We then summarized 4 conditions that are equivalent to a finite extension K/F being a Galois extension.

Reading: We started this lecture by stating Theorem 9 in Section 14.2.  We did not define linear characters, which play an important role in the proof, or state Theorem 7 or Corollary 8.  We will come back to this in Lecture 21.  We then followed Section 14.2 pretty closely up until the statement of the Fundamental Theorem of Galois Theory (pages 572-574).