1 Definitions

Let R be any Dedekind domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K. Fix a finite dimensional field extension L/K and let S denote the integral closureMathworldPlanetmath of R in L. For any basis x1,,xn of L over K, the determinant


whose entries are the trace of xixj over all pairs i,j, is called the discriminantPlanetmathPlanetmathPlanetmathPlanetmath of the basis x1,,xn. The ideal in R generated by all discriminants of the form


is called the discriminant ideal of S over R, and denoted Δ(S/R).

In the special case where S is a free R–module, the discriminant ideal Δ(S/R) is always a principal idealMathworldPlanetmath, generated by any discriminant of the form Δ(x1,,xn) where x1,,xn is a basis for S as an R–module. In particular, this situation holds whenever K and L are number fieldsMathworldPlanetmath.

2 Alternative notations

The discriminant is sometimes denoted with disc instead of Δ. In the context of number fields, one often writes disc(L/K) for disc(𝒪L/𝒪K) where 𝒪L and 𝒪K are the rings of algebraic integers of L and K. If K or 𝒪K is omitted, it is typically assumed to be or .

3 Properties

The discriminant is so named because it allows one to determine which ideals of R are ramified in S. Specifically, the prime idealsMathworldPlanetmathPlanetmath of R that ramify in S are precisely the ones that contain the discriminant ideal Δ(S/R). In the case R=, a theorem of Minkowski (http://planetmath.org/MinkowskisConstant) that any ring of integers S of a number field larger than has discriminant strictly smaller than itself, and this fact combined with the previous result shows that any number field K admits at least one ramified prime over .

4 Other types of discriminants

In the special case where L=K[x] is a primitive separable field extension of degree n, the discriminant Δ(1,x,,xn-1) is equal to the polynomial discriminant (http://planetmath.org/PolynomialDiscriminant) of the minimal polynomial f(X) of x over K[X].

The discriminant of an elliptic curveMathworldPlanetmath can be obtained by taking the polynomialPlanetmathPlanetmath discrimiant of its Weierstrass polynomial, and the modular discriminantMathworldPlanetmath of a complex lattice equals the discriminant of the elliptic curve represented by the corresponding lattice quotient.

Title discriminant
Canonical name Discriminant1
Date of creation 2013-03-22 12:37:57
Last modified on 2013-03-22 12:37:57
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 11R29
Related topic IntegralBasis
Related topic PolynomialDiscriminant
Related topic ModularDiscriminant
Defines discriminant ideal