 Solution

Suppose a rational function $P(t)/Q(t)\in {\mathbb{F}}_p(t)$, with nonzero polynomials $P,Q$, had $p$th power $t$. Then $P(t{)}^p=tQ(t{)}^p$, so comparison of degrees would give $p\mathrm{deg}P=1+p\mathrm{deg}Q$, impossible modulo $p$. Thus $X^p-t$ has no root in $K$, and part (b) proves it is irreducible.

In a splitting field containing $\alpha$ with ${\alpha }^p=t$,It has one distinct root with multiplicity $p$, so it is inseparable. This is the purely inseparable polynomial over a rational function field.