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.