An exercise in Jacobson's Basic Algebra 1 asks to prove that if $F$ is a field of characteristic $p$, then for any irreducible polynomial over $F$, its roots in splitting field occur with same multiplicity.
If we take finite field, then the roots of irreducible polynomial will occur with multiplicity $1$ in splitting field.
If we take $\mathbb{F}_p(t)$, then the irreducible polynomial $x^{p}-t$ will have single root with multiplicity $p$.
How can we construct examples of irreducible polynomials over fields of characteristic $p$, whose some roots are distinct and occur with multiplicity $>1$ in splitting field?
Let $F=\mathbb{F}_p(t)$. Take any separable polynomial $g\in F[X]$ which is $\pi$-Eisenstein for some irreducible $\pi\in \mathbb{F}_p[t].$ In particular, $g$ is irreducible.
For example, $g=X^2+tX+t$ does the job.
Then set $f=g(X^{p^k})$ for some arbitrary $k\geq 1$. By construction, $f$ is still $\pi$-Eisenstein, hence irreducible.
Now $f$ has $\deg(g)$ roots, all of multiplicity $p^k$.