Definition 15.1 (Infinite Galois definitions).

• $L∕K$ is separable if $\forall \alpha \in L$, the minimal polynomial $f_{\alpha }(X)\in K[X]$ for $\alpha$ is separable.

• $L∕K$ is normal if $f_{\alpha }(X)$ splits in $L$ for all $\alpha \in L$.

• $L∕K$ is Galois if it is separable and normal. Write $\mathrm{Gal}(L∕K):={\mathrm{Aut}}_K(L)$ in this case. If $L∕K$ is a finite Galois extension, then we have a Galois correspondence:

  $$\begin{array}{llll}\hfill \{\text{subextensions }K\subseteq K^{\prime }\subseteq L\}& \leftrightarrow \{\text{subgroups of }\mathrm{Gal}(L∕K)\}& \hfill & \\ \hfill K^{\prime }& \mapsto \mathrm{Gal}(K∕K^{\prime })& \hfill & \end{array}$$