TITLE: Prove that $K$ is algebraically closed
QUESTION [2 upvotes]: Let $K$ be a field of characteristic zero and $p$ a prime number such that $p^2$ divide at degree of all irreducible polynomial not linear in $K[x]$. Prove that $K$ is algebraically closed.
Thank you for some hints.

REPLY [0 votes]: First step: Let $f(x)$ be a nonconstant irreducible polynomial, let $L/K$ be a splitting field for $f(x)$, and let $G$ be the Galois group of $L/K$. Let $H \subset G$ be a Sylow-$p$ subgroup. What kind of intermediate field extension $K \subset L^H \subset L$ must $H$ correspond to? Assuming your Galois theory is strong enough, you should be able to conclude that $G$ must be a $p$-group.
Second step: Let $H \subset G$ be a subgroup of index $p$, which you should be able to show exists from the fact that $G$ is a $p$-group. Now describe $K \subset L^H \subset L$, and obtain a contradiction.
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