= Discriminant
{wiki}
For a degree-$n$ <polynomial> $p(X)=a_n\prod_{i=1}^n(X-r_i)$ over a <field>, with $a_n\ne0$ and <roots> in an <algebraic closure>, its discriminant is
$$
\operatorname{Disc}(p)=a_n^{2n-2}\prod_{i<j}(r_i-r_j)^2.
$$
It is a <polynomial> with <integer> coefficients in the coefficients of $p$, and lies in the original coefficient <field>. Its vanishing is equivalent to a repeated <root of a polynomial> in an <algebraic closure>, even when the coefficient <field> itself contains no <roots>. This detects separability without solving the <polynomial equation>.<Fields and Galois Theory by James S. Milne>[Chapter 4, Discriminants, and the resultant appendix]{ref}
For a <quadratic polynomial>, the expression is the <quadratic discriminant> $b^2-4ac$. For the <cubic polynomial> $X^3+bX+c$, it is $-4b^3-27c^2$. Thus $X^3-2$ has discriminant $-108$ over $\mathbb Q$, but after reduction to <characteristic> three its discriminant is zero and the resulting <polynomial> has a repeated <root> in its <splitting field>. <Discriminants> also indicate exceptional parameter values where distinct <polynomial roots> collide.
Back to article page
