§ Separable Extension is contained in Galois extension
- Recall that an extension is galois if it is separable and normal.
- Consider some separable extension .
- By primitive element, can be written as
- Since is separable, the minimal polynomial of , is separable, and so splits into linear factors.
- Build the splitting field of . This will contain , as where are the roots of .
- This is normal (since it is the splitting field of a polynomial).
- This is separable, since it is generated by separable elements , , , and so on.