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.