Section outline

  • We defined maps \(\mathcal{G}\) and \(\mathcal{F}\) between subfields of a field \(K\) and subgroups of \(\mathrm{Aut}(K)\). We proved properties of the maps. In particular, we proved that the maps define inclusion an reversing correspondence between subfields of \(K\) of the form \(\mathcal{F}(H)\) for some subgroup \(H \leq \mathrm{Aut}(K)\) and the subgroups of \(\mathrm{Aut}(K)\) of the form \(\mathcal{G}(L)\) for some subfield of \(K\).  

    We proved the Fundamental Theorem of Galois Theory and saw examples.

    Reading: section 2.4, pages 23-26 from the notes.