Example. $x^p - a$ [1705]

If $F$ is a field of characteristic $p$ and $a \in F$, then $x^{p} - a$ is inseparable. To prove that, we divide it into two cases:

1. If $a$ is a $p$-th power in $F$, let $\alpha \in \overline{F}$ such that $\alpha^{p} = a$, then $\alpha \in F$.Now $x^{p} - a = x^{p} - \alpha^{p} = (x - \alpha)^{p}$, $\alpha$ has multiplicity $p > 1$. So just by definition, $x^{p} - a$ is inseparable.
2. If $a$ is NOT a $p$-th power in $F$, then $\alpha \not\in F$. By the HW problem in section 1.4, $m_{\alpha, F}(x) = x^{p} - a$. Now taking the derivative of $x^{p} - a$ gives $px^{p-1} = 0$ and then by part 3 of Proposition 1704, $x^{p} - a$ is inseparable.

It follows that $F(\alpha) / F$ is inseparable in both cases.

Consider $F = \mathbb{F}_{p}(t) = \left\{ \frac{a(t)}{b(t)} \mid a(t), b(t) \in \mathbb{F}_{p}[t], b(t) \neq 0 \right\}$. $t \in F$ is not a $p$-th power in $F$ since if $t = \left( \frac{a(t)}{b(t)} \right)^{p}$ for some $a(t), b(t) \in \mathbb{F}_{p}[t]$ with $b(t) \neq 0$, then we would have $$ t \cdot \left( b(t) \right)^{p} = \left( a(t) \right)^{p}. $$ Suppose $\deg a(t) = m, \deg b(t) = n$. Then comparing degrees of both sides gives $np + 1 = mp \implies 1 = p(m - n)$. Since $p, m, n$ are integers, this is impossible. Therefore, $t$ cannot be a $p$-th power in $F$.

So again we have $x^{p} - t$ is irreducible in $F[x]$ and $F[x] / (x^{p} - t)$ is inseparable.