Finite field extensions

Show that an extension $F \supset P$ of prime degree has no proper $(\neq P, F)$ subfields.

Find a primitive element of the extension $\mathbb {Q}(\sqrt{p}, \sqrt{q})$, where $p$ and $q$ are prime numbers.

Find the dimension over $\mathbb {Q}$ of the splitting field of the polynomial $X^{p}-2$.

Show that over a field $P$ of characteristic $p>0$, for the polynomial $X^{p}-a$ there are only two possibilities: to be irreducible, or to be the $p$-th power of a linear polynomial.

Let $Z_{p}(Y)$ be the field of rational fractions of characteristic $p$. Show that $X^{p}-Y$ is a polynomial irreducible over $Z_{p}(Y)$, all of whose roots coincide.