Table of contents

1 Discriminant of a polynomial 2 Discriminant of a conic 3 Case of a quadratic form 4 Discriminant of an algebraic number field

Discriminant of a polynomial

The discriminant of a polynomial is a number which can be easily computed from the coefficients of the polynomial and which is zero if and only if the polynomial has a multiple root. For instance, the discrimant of the polynomial ax² + bx + c is b² - 4ac.

For the general definition, suppose

p(x) = xⁿ + a_n-1x^n-1 + ... + a₁x + a₀

 1     an-1     an-2      .         .        .    a0       0        .   .   .   0
 0     1        an-1     an-2       .        .    .       a0        0   .   .   0
 0     0        1        an-1     an-2       .    .       .        a0   0   .   0
 .     .        .        .         .        .    .
 .     .        .        .         .        .    .
 0     0        0        0         0        1    an-1    an-2       .   .   .  a0
 n  (n-1)an-1 (n-2)an-2   .         .       1a1   0        0        .   .   .   0
 0     n      (n-1)an-1 (n-2)an-2   .        .   1a1       0        0   .   .   0
 0     0        n       (n-1)an-1 (n-2)an-2  .    .       1a1       0   0   .   0
 .     .        .        .         .        .    .
 .     .        .        .         .        .    .
 0     0        0        0         0        n  (n-1)an-1  an-2      .   .  1a1  0
 0     0        0        0         0        0    n      (n-1)an-1  an-2 .   .  1a1

In the case n=4, this discriminant looks like this:

The discriminant of p(x) is thus equal to the resultant of p(x) and p'(x).

One can show that, up to sign, the discriminant is equal to

Π_i<j (r_i - r_j)²

p(x) = (x - r₁) (x - r₂) ... (x - r_n)

In order to compute discriminants, one does not evaluate the above determinant each time for different coefficient, but instead one evaluates it only once for general coefficients to get an easy-to-use formula. For instance, the discriminant of a polynomial of third degree is a₁²a₂² - 4a₀a₂³ -4a₁³ + 18 a₀a₁a₂ - 27a₀².

The discriminant can be defined for polynomials over arbitrary fieldss, in exactly the same fashion as above. The product formula involving the roots r_i remains valid; the roots have to be taken in some splitting field of the polynomial.

Discriminant of a conic

For conic section defined by real polynomials of the form

Ax²+Bxy+Cy²+Dx+Ey+F=0,

B²-4AC,

Case of a quadratic form

There is a substantive generalisation, to quadratic forms Q over any field K of characteristic ≠ 2. These can be written as a sum of terms

a_iL_i²

Discriminant of an algebraic number field

If K is an algebraic number field and R its ring of integers, the discriminant of K is associated to R and in some sense measures how large R is. In the special case of R = Z[α] for some algebraic integer α in K, it is simple to define, as the discriminant of the minimal polynomial P_α of α. This suffices, for example, in the case of the Gaussian integers: we take P(T) = T² + 1 for the choice α = i and calculate the discriminant as −4.

This in fact works for any quadratic field or cyclotomic field; but certainly not in general. There we can only be sure that Z[α] can be chosen, of finite index in R as abelian group. This gives a factor (of the index) which is awkward to apply. The correct definition comes through a recognition that the discriminant of a polynomial is a square of a Vandermonde determinant, and that determinant is what we should generalise. The analogue in the general case is this: let the ω_i be an integral basis (i.e. basis for R as Z-module) and form

det(ω_i^(j))

Why this is the correct approach is best studied in terms of the real vector space

There is also a formula for the discriminant related to the quadratic form definition above, starting from the field trace. In the theory of Pontryagin duality for completions of K as [[local field]s, the related different ideal occurs naturally in matching up Haar measures. This accounts for the role of the discriminant in the functional equation for the Dedekind zeta function, and thence in the analytic class number formula, and Brauer-Siegel theorem.