Irreducible $f(x) \in F[x]$ of prime degree, $E/F$ finite extension, $p \mid [E:F]$.

Question

Let $F$ be a field and let $f(x) \in F[x]$ be irreducible of prime degree $p$. Let $E/F$ be a finite extension. Prove: If $f(x)$ is not irreducible in $E[x]$, then $p \mid [E:F]$. (Hint: Consider a field $L$ with $E \subseteq L$ and $L$ as a root of $\alpha$ of $f(x)$.)

Proof: Let $F$ be a field and let $f(x) \in F[x]$ be irreducible of prime degree $p$. Let $E/F$ be a finite extension. By Proposition 20 there exists a field $L$ containing $F$ with $[L:F] = p$ in which $f(x)$ has a root $\alpha$. If $E \subseteq L$ then we can consider $p = [L : F] = [L : E][E : F]$. It must follow that $[E:F] = p$ since if $[L:E] = p$, then $[E:F]=1$ which would contradict $f(x)$ being irreducible in $F$. Hence $[E:F] = p$ and immediately $p \mid [E:F]$. Now if $L \subseteq E$ then we consider $[E:F] = [E:L][L:F] = [E:L]p$ and thus $p \mid [E:F]$.

Proposition 20: Let $F$ be a field. Let $p(x) \in F[x]$ be irreducible of degree $n$. Then there exists a field $K$ containing $F$ with $[K:F] = n$ in which $p(x)$ has a root.

Is this proof correct? If not, what is wrong and how would I fix it. If its right, alternate proofs/approaches welcomed.

Answer 1

Your proof technique doesn't quite work. There's no reason that $L$ needs to be comparable to $E$. The hint I believe is suggesting the following approach. If $f(x)$ takes a root $\alpha \in E$ we are done (why?). Otherwise let $g(x)$ be an irreducible factor of $f(x)$ over $E$. Now apply Proposition 20 to $E$ and $g(x)$. A bit more clever degree arithmetic should finish the argument now.

Answer 2

Let $L$ be the splitting field of $f(x)$ over $\mathbb F$, and $K$ the splitting field of $f(x)$ over $E$. Then $L\subseteq K$, and $p\mid [L:F]$, so $p\mid[K:F]$. But $f$ is reducible over $E$, this means that $p$ does not divide $[K:E]$. Hence $p\mid[E:F]$. Q.E.D.

Answer 3

Let $F$ be a field, $f(x) \in F[x]$ irreducible of prime degree $p$. Let $E/F$ be a finite extension. Suppose $f(x)$ is reducible in $E[x]$. Let $\alpha$ be a root of $f(x)$. Consider $F(\alpha)$. It follows that $[F(\alpha) : F] = p = \deg \alpha$ since $f(x)$ is irreducible in $F$ of degree $p$. Now observe that $[E(\alpha) : E] < p$ since $f(x)$ is reducible in $E$ and $f(x)$ is not the minimal polynomial for $\alpha$ over $E$, but $m_{\alpha, E}(x) \mid f(x)$ so $[E(\alpha) : E] = \deg m_{\alpha, E}(x) < \deg f(x) = p$. Now \begin{align*} [E(\alpha) : F] &= [E(\alpha) : F(\alpha)][F(\alpha) : F] = p[E(\alpha) : F(\alpha)]\\ [E(\alpha) : F] &= [E(\alpha) : E][E : F] \end{align*} So $ [E(\alpha) : E][E : F] = p[E(\alpha) : F(\alpha)]$. It follows that $p \mid [E(\alpha) : E][E : F]$. It immediately follows that $p \mid [E(\alpha) : E]$ or $p \mid [E : F] $. Conclude that $p \mid [E:F]$.