= Solution
In the distinct-root convention, a <polynomial> is separable if it has no repeated roots in its <splitting field>, equivalently $\gcd(f,f')=1$. A common factorwise convention instead requires each distinct irreducible factor to have no repeated roots; repeated occurrences of an already separable factor do not affect the <splitting field>. Under either separability convention, the <splitting field> is a finite normal separable extension and
$$
\boxed{|\operatorname{Gal}(L/K)|=[L:K].}
$$
For failure in the inseparable case, take $K=\mathbb F_p(t)$ and $f=X^p-t$. It is irreducible, for example by Eisenstein at the prime $t$ in $\mathbb F_p[t]$. Its <splitting field> is $L=K(t^{1/p})$ of degree $p$, but the <polynomial> has just one distinct root. Every $K$-<automorphism> fixes that root, so the <Galois group> is trivial rather than having order $p$.
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