TITLE: Irreducibility of the polynomial $f (x) = x^{2p} − x ^p + t$
QUESTION [5 upvotes]: If $k$ is a ﬁeld of characteristic $p > 0$ and $f (x) = x^{2p} − x ^p + t \in k(t)[x]$, how can we show  that $f (x)$ is an irreducible polynomial in $k(t)[x]$ and that $f (x)$ is inseparable?
If $f(x)$ is irreducible then $D_xf(x) = 0 \implies( f, D_xf) = ( f, 0) = f$. Hence, if $f(x)$ is irreducible then $( f, D_xf) \neq 1 \implies f(x)$ is inseparable.

REPLY [2 votes]: $K$ is a field $\implies K[t]$ is a U.F.D.
By gauss lemma: $f(x)$ is irreducible in $K(t)[x] \iff f(x)$ is irreducible in $K[t][x]$
then $K[t][x]= K[x][t]$ 
$f(x)$ is irreducible in $K[x][t]$ being a linear polynomial in $K[x][t]$ 
hence $f(x)$ is irreducible in $K(t)[x] $