= Solution
{c}
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 $p\deg P=1+p\deg Q$, 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 $\alpha^p=t$,
$$
\boxed{X^p-t=(X-\alpha)^p.}
$$
It has one distinct <root> with multiplicity $p$, so it is inseparable. This is the <purely inseparable polynomial over a rational function field>.
Back to article page
