Every nonzero polynomial over a field has a splitting field

Let $P(n)$ assert the theorem for every field and every nonzero polynomial of degree $n$. We prove $P(n)$ for all $n$ by [F1].

If $n=0$, the polynomial is a nonzero constant. It splits over $F$, and its empty root set generates $F$, so $F$ itself is a splitting field.

Let $n\ge 1$ and assume $P(m)$ for every $m<n$. By [F2], choose a root $\alpha$, put $K=F(\alpha )$, and choose $g\in K[x]$ with $f=(x-\alpha )g$ and $\mathrm{deg}g=n-1<n$.

The induction hypothesis over the field $K$ gives a splitting field $L\mathrm{/}K$ of $g$. Then $f$ splits over $L$ by [F2], and $L=K(\text{roots of }g)=F(\alpha ,\text{roots of }g)$ is generated over $F$ by roots of $f$. Thus $L\mathrm{/}F$ is a splitting field of $f$.

Steps 1.2, 1.3, and 2.1 verify the strong-induction hypothesis at every $n$, so [F1] proves the theorem.