Discriminant

In mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the x-axis). This generalises to polynomials of any degree the case of a quadratic polynomial ax² + bx + c, for which the discriminant is b² − 4ac, the quantity under the square root sign in the formula for the roots (or quadratic formula). Discriminants in algebraic number theory are closely related, and contain information about ramification. In fact the more geometric types of ramification are also related to more abstract types of discriminant, making this a central algebraic idea in many applications.

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 discriminant of the polynomial ax² + bx + c is b² − 4ac.

For the general definition, suppose

p(x) = xⁿ + a_n−1xⁿ⁻¹ + ... + a₁x + a₀

is a polynomial with real coefficients. The discriminant of this polynomial is defined as the determinant of the (2n − 1)×(2n − 1) matrix

 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:

<math>\begin{vmatrix}

& 1 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \\
& 0 & 1 & a_3 & a_2 & a_1 & a_0 & 0 \\
& 0 & 0 & 1 & a_3 & a_2 & a_1 & a_0 \\
& 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \\
& 0 & 4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \\
& 0 & 0 & 4 & 3a_3 & 2a_2 & 1a_1&  0 \\
& 0 & 0 & 0 & 4 & 3a_3 & 2a_2 & 1a_1 \\

\end{vmatrix}<math>

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

where r₁, ..., r_n are the (complex) numbers such that

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

Therefore, p has a multiple root if and only if the discriminant is zero. Note however that this multiple root can be complex.

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 fields, 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,

the discriminant is equal to

B² − 4AC,

and determines the shape of the conic section. If the discriminant is less than 0, the equation is of an ellipse or a circle. If the discriminant equals 0, the equation is that of a parabola. If the discriminant is greater than 0, the equation is that of a hyperbola. This formula will not work for degenerate cases (when the polynomial factorises).

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

where the L_i are linear forms and 1 ≤ i ≤ n where n is the number of variables. Then the discriminant is the product of the a_i, taken in K/K², and is then well-defined (i.e., up to squares). A more invariant way to say this is as (the class of) the determinant of a symmetric matrix for Q.

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))

where the superscripts mean that we take the conjugates. This (squared) leads to the correct general definition.

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

<math>K \otimes \mathcal R\,<math>

and the embedding into it of R as a lattice. The determinant involved in the discriminant then has a simple interpretation as a volume of a fundamental region for R.

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 fields, 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.

A theorem of Stickelberger states that the discriminant D of an algebraic number field must be congruent to 0 or 1 modulo 4. A result of Kronecker is that D = 1 is possible only for the rational number field; this entails that every other number field has some ramified prime p in it. Lower bounds for discriminants, in terms of the degree, are proved by methods from the geometry of numbers and analytic number theory. In the case of an abelian extension, one of the results of class field theory (conductor-discriminant formula) is a factorisation of the discriminant according to characters, which in the case of an abelian extension of Q are Dirichlet characters.