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.