# $E/F, K/F$ fields with $E\cap K = F$, $E/F$ Galois, $\alpha \in K$ with min poly $f(x) \in F[x]$ then $f$ irreducible over $E$.

Let $$E/F$$ and $$K/F$$ be field extensions contained in some common field $$L$$. Suppose $$E/F$$ is Galois and that $$E \cap K = F$$. Let $$\alpha \in K$$ be algebraic over $$F$$ with minimal polynomial $$f$$. Show that $$f$$ is irreducible over $$E$$.

So far: I know that $$f$$ has no roots in $$E$$, since otherwise it would have all roots in $$E$$ by normality, whence $$\alpha \in E$$ (contradicting $$E \cap K = F$$). But I don't see why $$f$$ mustn't split in $$E$$ into irreducible factors.

• Do you perchance mean "common field $F$"? – Robert Lewis May 8 '19 at 0:30
• @RobertLewis I mean that $E, K \subset L$. – Freddie May 8 '19 at 0:31
• OK, I get it now. Cheers! – Robert Lewis May 8 '19 at 0:31