## Tag Archives: discriminant

### 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.