Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / A / A2 / a / Solution

A splitting field of $f$ over $F$ is an extension field in which $f$ factors into linear factors and which is generated over $F$ by those roots. The generation requirement excludes unnecessary extra elements.

For existence, induct on the degree. An irreducible factor $p$ of $f$ gives a field $F_1=F[X]/(p)$, since an irreducible polynomial generates a maximal ideal in the polynomial ring over a field. The class $\alpha$ of $X$ is a root of $p$ and hence of $f$. By the factor theorem, $f=(X-\alpha )g$ over $F_1$, with $\mathrm{deg}g=\mathrm{deg}f-1$. Induction gives a splitting field of $g$ over $F_1$. It contains all roots of $f$; taking the subfield generated by these roots over $F$ gives exactly a splitting field of $f$. The degree-one case is immediate. Thus