Let and let with discriminant . Show that has multiple roots, three distinct real roots, or one real root and two non-real roots according as , , .

has multiple roots if and only if .

Supoose . Since cubic equation always has at least one root in , assume , . Then Galois group of is equal to that of over , which is the symmetric group or the alternating group .

if and only if is contained in symmetric group . Since has only two cases. if and only if is which means has three distinct roots in

Correspondingly, we get if and only if one real root and two non-real roots.

Jacobson p260.