In Dummit and Foote's book,there is a Proposition
Let $K_1$ and $K_2$ be Galois extensions of a field $F$.Then the intersection $K_1\cap K_2$ is Galois over $F$. The proof is as following:
Suppose $p(x)$ is an irreducible polynomial in $F[x]$ with a root $a$ in $K_1\cap K_2$.Since $a\in K_1$ and $K_1/F$ is Galois,all the roots of $p(x)$ lie in $K_1$.Similarly,all the roots lie in $K_2$,hence all the roots of $p(x)$ lie in $K_1\cap K_2$.It follows easily that $K_1\cap K_2$ is Galois as in Theorem 13.
I feel a little confused about the last step,By Pro 13($K/F$ is Galois iff K is the spltting field of some separable polynomial over $F$),we need to show find a separable polynomial.But $p(x)$ mentioned above is irreducible,if the $Char(F)\neq 0$,irreducible polynomial may not be separable.
Can anyone help me?Thanks in advance!