# 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 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.