discriminant

1 Definitions

Let $R$ be any Dedekind domain with field of fractions $K$. Fix a finite dimensional field extension $L/K$ and let $S$ denote the integral closure of $R$ in $L$. For any basis $x_1,\mathrm{\dots },x_n$ of $L$ over $K$, the determinant

$$\mathrm{\Delta }(x_1,\mathrm{\dots },x_n):=det[\mathrm{Tr}(x_i{x}_j)],$$

whose entries are the trace of $x_i{x}_j$ over all pairs $i,j$, is called the discriminant of the basis $x_1,\mathrm{\dots },x_n$. The ideal in $R$ generated by all discriminants of the form

$$\mathrm{\Delta }(x_1,\mathrm{\dots },x_n),x_i\in S$$

is called the discriminant ideal of $S$ over $R$, and denoted $\mathrm{\Delta }(S/R)$.

In the special case where $S$ is a free $R$–module, the discriminant ideal $\mathrm{\Delta }(S/R)$ is always a principal ideal, generated by any discriminant of the form $\mathrm{\Delta }(x_1,\mathrm{\dots },x_n)$ where $x_1,\mathrm{\dots },x_n$ is a basis for $S$ as an $R$–module. In particular, this situation holds whenever $K$ and $L$ are number fields.

2 Alternative notations

The discriminant is sometimes denoted with $\mathrm{disc}$ instead of $\mathrm{\Delta }$. In the context of number fields, one often writes $\mathrm{disc}(L/K)$ for $\mathrm{disc}({𝒪}_L/{𝒪}_K)$ where ${𝒪}_L$ and ${𝒪}_K$ are the rings of algebraic integers of $L$ and $K$. If $K$ or ${𝒪}_K$ is omitted, it is typically assumed to be $\mathbb{Q}$ or $\mathbb{Z}$.

3 Properties

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

4 Other types of discriminants

In the special case where $L=K[x]$ is a primitive separable field extension of degree $n$, the discriminant $\mathrm{\Delta }(1,x,\mathrm{\dots },x^{n-1})$ 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 curve can be obtained by taking the polynomial discrimiant of its Weierstrass polynomial, and the modular discriminant 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