# Different ideal

In algebraic number theory, the **different ideal** is an invariant attached to an extension of algebraic number fields.

Let *L*/*K* be such an extension, with rings of integers *O*_{L} and *O*_{K} respectively. The relative trace defines a bilinear form on *L* by

in which the dual of *O*_{L} is a fractional ideal of *L* containing *O*_{L}. The *(relative) different* δ_{L/K} is the inverse of this fractional ideal: it is an ideal of *O*_{L}.

The relative norm of the relative different is equal to the relative discriminant Δ_{L/K}. In a tower of fields *L*/*K*/*F* the relative differents are related by δ_{L/F} = δ_{L/K} δ_{K/F}.

## Ramification

The relative different encodes the ramification data of the field extension *L*/*K*. A prime ideal *p* of *K* ramifies in *L* if and only if it divides the relative discriminant Δ_{L/K}. If

*p*=*P*_{1}^{e(1)}...*P*_{k}^{e(k)}

is the factorisation of *p* into prime ideals of *L* then *P*_{i} divides the relative different δ_{L/K} if and only if *P*_{i} is ramified, that is, if and only if the ramification index *e(i)* is greater than 1. The precise exponent to which a ramified prime *P* divides δ is termed the **differential exponent** of **P** and is equal to *e*-1 if *P* is tamely ramified: that is, when *P* does not divide *e*. In the case when *P* is wildly ramified the differential exponent lies in the range *e* to *e*+ν_{P}(e)-1.

## Local computation

The different may be defined for an extension of local fields *L*/*K*. In this case we may take the extension to be simple, generated by a primitive element α which also generates a power integral basis. If *f* is the minimal polynomial for α then the different is generated by *f'*(α).