discriminant of a polynomial

The discriminant is a single number, computed from a polynomial's coefficients, that detects whether the polynomial has a repeated root. It generalizes the familiar b^2 - 4ac from the quadratic formula: when it vanishes, two roots have collided; when it is nonzero, all the roots are distinct.

Formally, for a degree-n polynomial f with roots r_1, ..., r_n (in a splitting field) and leading coefficient a_n, the discriminant is disc(f) = a_n^{2n-2} times the product over all i < j of (r_i - r_j)^2. Because it is a symmetric function of the roots, the fundamental theorem of symmetric functions guarantees it is a polynomial in the coefficients of f, hence computable without ever finding the roots. It vanishes exactly when r_i = r_j for some i != j, i.e. when f has a repeated root, equivalently when gcd(f, f') is nontrivial.

In Galois theory the discriminant carries group-theoretic information. The square root of the discriminant, the product of (r_i - r_j) over i < j, is fixed by exactly the even permutations of the roots. Hence for a separable polynomial, the Galois group (viewed as a permutation group on the roots) lies inside the alternating group A_n if and only if disc(f) is a square in the base field. For a cubic this is precisely the test distinguishing Galois group A_3 from S_3.