# Difference between revisions of "Discriminant of an algebraic number field"

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In [[algebraic number theory]], the '''discriminant of an algebraic number field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric structure of the [[ring of integers]] and also encodes [[ramification]] data. | In [[algebraic number theory]], the '''discriminant of an algebraic number field''' is an invariant attached to an [[field extension|extension]] of [[algebraic number field]]s which describes the geometric structure of the [[ring of integers]] and also encodes [[ramification]] data. | ||

## Latest revision as of 06:20, 18 February 2009

In algebraic number theory, the **discriminant of an algebraic number field** is an invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and also encodes ramification data.

The *relative discriminant* Δ_{K/L} is attached to an extension *K* over *L*; the *absolute discriminant* of *K* refers to the case when *L* = **Q**.

## Absolute discriminant

Let *K* be a number field of degree *n* over **Q**. Let *O*_{K} denote the ring of integers or maximal order of *K*. As a free **Z**-module it has a rank *n*; take a **Z**-basis . The discriminant

Since any two **Z**-bases are related by a unimodular change of basis, the discriminant is independent of the choice of basis.

An alternative definition makes use of the *n* different embeddings of *K* into the field of complex numbers **C**, say σ_{1}, ...,σ_{n}:

We see that these definitions are equivalent by observing that if

then

and then taking determinants.