Tag Archives: cubic equation

The discriminant of a cubic equation and its relation with roots

\mathbf{Problem:} Let F=\mathbb{R} and let f(x)=x^3-a_1x^2+a_2x-a_3 with discriminant d. Show that f(x) has multiple roots, three distinct real roots, or one real root and two non-real roots according as d=0, d>0, d<0.

\mathbf{Proof:} f has multiple roots if and only if d=0.

Supoose d\neq 0. Since cubic equation always has at least one  root in \mathbb{R}, assume f(x)=(x-r)g(x), r\in \mathbb{R}. Then Galois group G_f of f is equal to that of g(x) over \mathbb{R}, which is the symmetric group S_2 or the alternating group A_2=1.

d>0 if and only if G_f is contained in symmetric group A_3. Since G_f has only two cases. d>0 if and only if G_f is A_2=1 which means f has three distinct roots in \mathbb{R}

Correspondingly, we get d<0 if and only if   one real root and two non-real roots.

\text{Q.E.D}\hfill \square

\mathbf{Remark:} Jacobson p260.