Let $K / F$ be a finite and separable extension of fields and let $L$ a normal closure of $K / F$. Can I state that $L / F$ is also finite and separable?
Asked
Active
Viewed 29 times
1
-
You may state it, and it would be true. Being separable and finite, the extension is simple: $K=F(\theta)$ for a suitable $\theta\in K$. Then adjoin the (finitely many) roots of $\theta$’s minimal $F$-polynomial to get $L$. – Lubin Aug 29 '18 at 03:35