= Solution
{c}
Let $\alpha$ be a <root> in a <splitting field>. The preceding characteristic-$p$ identity gives $X^p-a=(X-\alpha)^p$, so there is only one distinct <root>. If the <polynomial> were reducible over $F$, choose a monic proper factor $g\in F[X]$ of <degree> $d$, with $1\leq d<p$. In the splitting field it must be $g=(X-\alpha)^d$. Its coefficient of $X^{d-1}$ is $-d\alpha$, which belongs to $F$. Since $d$ is nonzero and thus invertible in the <field> $F$, this forces $\alpha\in F$, contradicting the absence of a <root> there. Therefore \b[$X^p-a$ is irreducible over $F$].
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