Section outline

  • We started to study the Galois group. 

    We proved that the following four statements, regarding a finite extension K/F, are equivalent:

    1. K/F is normal and separable.
    2. K is the splitting field of a set of separable polynomials over F.
    3. |\mathrm{Gal}(K/F)| = [K : F].  
    4. F = \mathcal{F}(\mathrm{Gal}(K/F)).

    An extension that satisfies one (and therefore all) of the above statements, is called a Galois extension.

    Reading: pages 17-22 from the notes.