Let $K \subset L$ be a finite Galois extension. Assume that there is $a \in L$ such that $\sigma(a) \not= a$ for every $\sigma \in$ Gal$(L/K)$. Let $P$ be the minimal polynomial of $a$ over $K$.
(a) Show that deg $P$ = |Gal $(L/K)$|.
(b) Show that $L = K(a).$
I can see how (b) will follow (a). Any help with the first part would be appreciated - I know that deg $P$ will divide|Gal $(L/K)$|. I tried looking at the roots of the minimal polynomial but did not get anywhere useful.