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

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Since any two '''Z'''-bases are related by a [[unimodular matrix|unimodular]] change of basis, the discriminant is independent of the choice of basis. | Since any two '''Z'''-bases are related by a [[unimodular matrix|unimodular]] change of basis, the discriminant is independent of the choice of basis. | ||

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+ | An alternative definition makes use of the ''n'' different embeddings of ''K'' into the field of [[complex number]]s '''C''', say σ<sub>1</sub>, ...,σ<sub>''n''</sub>: | ||

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+ | :<math>\Delta_K = (\det \sigma_i(\omega_j) )^2 .\,</math> | ||

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+ | We see that these definitions are equivalent by observing that if | ||

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+ | :<math>A = \left(\sigma_i(\omega_j) \right) \,</math> | ||

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+ | then | ||

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+ | :<math>A^\top A = \left( \sum_j \sigma_j(\omega_i) \sigma_j(\omega_k) \right) = \left(\operatorname{tr}(\omega_i\omega_k) \right) ,\,</math> | ||

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+ | and then taking determinants. |

## Revision as of 20:28, 23 December 2008

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.