Let \(k\) be a field and \(f\in k[x]\) a nonzero polynomial. A splitting field of \(f\) over \(k\) is a field extension \(F/k\) such that \(f\) splits into linear factors in \(F[x]\) and \(f\) does not split into linear factors in \(E[x]\) for any proper subfield \(E\subsetneq F\) containing \(k\).
Equivalently, \(f\) splits in \(F[x]\) as \(f=c\prod_{i=1}^n(x-\alpha_i)\) and \(F=k(\alpha_1,\ldots,\alpha_n)\): a subfield of \(F\) containing \(k\) over which \(f\) splits contains every root \(\alpha_i\), and \(k(\alpha_1,\ldots,\alpha_n)\) is such a subfield.
What is the splitting field of a polynomial $f \in k[x]$?