9 March - 15 March

We proved that any two splitting fields, \(K, K'\), of a polynomial in \(f \in F[x]\), over \(F\) are isomorphic. More generally, if \(\sigma : F \rightarrow F'\) is an isomorphism, \(f \in F[x]\), and \(K, K'\) are splitting fields of \(f\) and \(\sigma^*(f)\) over \(F, F'\) respectively, then \(\sigma\) can be extended to \(\tau : K \rightarrow K'\). If \(\alpha \in K\) is a root of \(f\), and \(\alpha' \in K'\) is a root of \(\sigma^*(\min(F, \alpha))\), then \(\tau\) can be chosen so that \(\tau(\alpha) = \alpha'\).

We discussed \(n\)-th roots of unity (not a comprehensive study of cyclotomic extensions, but rather a reminder of basic facts). 

We defined normal extensions and proved that if \(K/F\) is normal if and only if any irreducible \(P \in F[x]\) that has a root in \(K\) splits in \(K\). We discussed the significance of normal extensions in the context of the Isomorphism Extension Theorem: an extension of \(\mathrm{id} : F \rightarrow F\) to \(K\) is not only an embedding of \(K\) to \(\overline{F}\), it is an automorphism of \(K\).

Reading: pages 12-13 from the notes.

Reading: section 2.1 form the notes.