If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed?

Question

Suppose $E/F$ is an algebraic extension, where every polynomial over $F$ has a root in $E$. It's not clear to me why $E$ is actually algebraically closed.

I attempted the following, but I don't think it's correct:

I let $f$ be an irreducible polynomial in $E[X]$. I let $\alpha$ be a root in some extension, so $f=m_{\alpha,E}$. Since $\alpha$ is algebraic over $E$, it is also algebraic over $F$, let $m_{\alpha,F}$ be it's minimal polynomial. I now let $K$ be a splitting field of $m_{\alpha,F}$, which is a finite extension since each root has finite degree over $F$.

If $m_{\alpha,F}$ is separable, then $K/F$ is also separable, so as a finite, separable extension, we can write $K=F(\beta)$ for some primitive element $\beta$. By assumption, $m_{\alpha,F}$ has a root in $E$, call it $r$. Then we can embed $F(\beta)$ into $r$ by mapping $\beta$ to $r$. It follows that $m_{\alpha,F}$ splits in $E$. Since $f\mid m_{\alpha,F}$, we must also have the $f$ is split in $E$.

But what happens if $m_{\alpha,F}$ is not separable? In such case, $F$ must have characteristic $p$. I know we can express $m_{\alpha,F}=g(X^{p^k})$ for some irreducible, separable polynomial $g(X)\in F[X]$. But I'm not sure what follows after that.

NB: I say $E$ is algebraically closed if every nonconstant polynomial in $E[X]$ has a root in $E$.

Answer 1

This is a recent qual question at my school. :)

If $F$ is not perfect, say it has characteristic $p$. Let $F_{perf}$ be the extension of $F$ obtained by adjoining a root for $x^{p^n}-a$ for every $n \in \mathbb N, a \in F$. Since these polynomials are purely inseparable, $F_{perf}$ embeds (via a unique isomorphism) into $E$, and we may identify $F_{perf}$ with its isomorphic copy in $E$.

Two facts for you to verify: $F_{perf}$ is a perfect field, and every element of $F_{perf}$ is a root of $x^{p^n}-a$ for some $n \in \mathbb N$ and $a \in F$.

Now let $f$ be a polynomial with coefficients in $F_{perf}$, say $f(x) = x^k + a_{k-1} x^{k-1} + \cdots + a_0$. Then for sufficiently large $n$, $a_i^{p^n} \in F$ for all $i$. So $$f^{p^n}(x) = x^{k p^n} + a_{k-1}^{p^n} x^{(k-1)p^n} + \cdots + a_0^{p^n} \in F[x],$$ which has a root in $E$ by assumption. But $f$ and $f^{p^n}$ have the same roots, hence $f$ has a root in $E$. So without loss of generality, we may assume $F = F_{perf}$, and Bruno's answer gets us the rest of the way.

Answer 2

Note: feel free to ignore the warzone in the comments; it's not really relevant anymore.

If $F$ is perfect, we can proceed like this. Let $f$ be a polynomial with coefficients in $F$. Let $K/F$ be a splitting field for $f$. Then $K=F(\alpha)$ for some $\alpha \in K$. Let $g$ be the minimal polynomial of $\alpha$ over $F$. Then $g$ has a root in $E$ by assumption, hence $E$ contains a copy of $F(\alpha)$, i.e. a splitting field for $f$.

Thus every $f\in F[X]$ splits in $E$. Now I claim that $E$ is algebraically closed. Let $E'/E$ be an algebraic extension and let $\beta \in E'$. By transitivity, $\beta$ is algebraic over $F$; let $h(X)$ be its minimal polynomial over $F$. By the above, $h$ splits in $E$, and therefore $\beta \in E$. Thus $E$ is algebraically closed.