Discriminant - Discriminant of A Quadratic Form

Discriminant of A Quadratic Form

There is a substantive generalization to quadratic forms Q over any field K of characteristic ≠ 2. For characteristic 2, the corresponding invariant is the Arf invariant.

Given a quadratic form Q, the discriminant or determinant is the determinant of a symmetric matrix S for Q.

Change of variables by a matrix A changes the matrix of the symmetric form by which has determinant so under change of variables, the discriminant changes by a non-zero square, and thus the class of the discriminant is well-defined in K/(K*)2, i.e., up to non-zero squares. See also quadratic residue.

Less intrinsically, by a theorem of Jacobi quadratic forms on can be expressed in diagonal form as

or more generally quadratic forms on V as a sum

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, which is well-defined as a class in K/(K*)2.

For K=R, the real numbers, (R*)2 is the positive real numbers (any positive number is a square of a non-zero number), and thus the quotient R/(R*)2 has three elements: positive, zero, and negative. This is a cruder invariant than signature (n₀, n₊, n₋), where n₀ is the number 0s and n_± is the number of ±1s in diagonal form. The discriminant is then zero if the form is degenerate, and otherwise it is the parity of the number of negative coefficients,

For K=C, the complex numbers, (C*)2 is the non-zero complex numbers (any complex number is a square), and thus the quotient C/(C*)2 has two elements: non-zero and zero.

This definition generalizes the discriminant of a quadratic polynomial, as the polynomial homogenizes to the quadratic form which has symmetric matrix

whose determinant is Up to a factor of −4, this is

The invariance of the class of the discriminant of a real form (positive, zero, or negative) corresponds to the corresponding conic section being an ellipse, parabola, or hyperbola.