# Different ideal

In algebraic number theory, the **different ideal** (sometimes simply the **different**) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field *K*, with respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882.[1][2]

## Definition

If *O*_{K} is the ring of integers of *K*, and *tr* denotes the field trace from *K* to the rational number field **Q**, then

is an integral quadratic form on *O*_{K}. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case *K* = **Q**). Define the *inverse different* or *codifferent*[3][4] or *Dedekind's complementary module*[5] as the set *I* of *x* ∈ *K* such that tr(*xy*) is an integer for all *y* in *O*_{K}, then *I* is a fractional ideal of *K* containing *O*_{K}. By definition, the **different ideal** δ_{K} is the inverse fractional ideal *I*^{−1}: it is an ideal of *O*_{K}.

The ideal norm of *δ*_{K} is equal to the ideal of *Z* generated by the field discriminant *D*_{K} of *K*.

The *different of an element* α of *K* with minimal polynomial *f* is defined to be δ(α) = *f*′(α) if α generates the field *K* (and zero otherwise):[6] we may write

where the α^{(i)} run over all the roots of the characteristic polynomial of α other than α itself.[7] The different ideal is generated by the differents of all integers α in *O*_{K}.[6][8] This is Dedekind's original definition.[9]

The different is also defined for a finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.

## Relative different

The **relative different** δ_{L / K} is defined in a similar manner for an extension of number fields *L* / *K*. The relative norm of the relative different is then equal to the relative discriminant Δ_{L / K}.[10] In a tower of fields *L* / *K* / *F* the relative differents are related by δ_{L / F} = δ_{L / K}*δ*_{K / F}.[5][11]

The relative different equals the annihilator of the relative Kähler differential module :[10][12]

The ideal class of the relative different δ_{L / K} is always a square in the class group of *O*_{L}, the ring of integers of *L*.[13] Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of *O*_{K}:[14] indeed, it is the square of the Steinitz class for *O*_{L} as a *O*_{K}-module.[15]

## Ramification

The relative different encodes the ramification data of the field extension *L* / *K*. A prime ideal *p* of *K* ramifies in *L* if the factorisation of *p* in *L* contains a prime of *L* to a power higher than 1: this occurs if and only if *p* divides the relative discriminant Δ_{L / K}. More precisely, 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.[11][16] 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*.[17] In the case when *P* is wildly ramified the differential exponent lies in the range *e* to *e* + ν_{P}(e) − 1.[16][18][19] The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[20]

## 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).

## Notes

- Dedekind 1882
- Bourbaki 1994, p. 102
- Serre 1979, p. 50
- Fröhlich & Taylor 1991, p. 125
- Neukirch 1999, p. 195
- Narkiewicz 1990, p. 160
- Hecke 1981, p. 116
- Hecke 1981, p. 121
- Neukirch 1999, pp. 197–198
- Neukirch 1999, p. 201
- Fröhlich & Taylor 1991, p. 126
- Serre 1979, p. 59
- Hecke 1981, pp. 234–236
- Narkiewicz 1990, p. 304
- Narkiewicz 1990, p. 401
- Neukirch 1999, pp. 199
- Narkiewicz 1990, p. 166
- Weiss 1976, p. 114
- Narkiewicz 1990, pp. 194,270
- Weiss 1976, p. 115

## References

- Bourbaki, Nicolas (1994).
*Elements of the history of mathematics*. Translated by Meldrum, John. Berlin: Springer-Verlag. ISBN 978-3-540-64767-6. MR 1290116. - Dedekind, Richard (1882), "Über die Discriminanten endlicher Körper",
*Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen*,**29**(2): 1–56. Retrieved 5 August 2009 - Fröhlich, Albrecht; Taylor, Martin (1991),
*Algebraic number theory*, Cambridge Studies in Advanced Mathematics,**27**, Cambridge University Press, ISBN 0-521-36664-X, Zbl 0744.11001 - Hecke, Erich (1981),
*Lectures on the theory of algebraic numbers*, Graduate Texts in Mathematics,**77**, translated by George U. Brauer; Jay R. Goldman; with the assistance of R. Kotzen, New York–Heidelberg–Berlin: Springer-Verlag, ISBN 3-540-90595-2, Zbl 0504.12001 - Narkiewicz, Władysław (1990),
*Elementary and analytic theory of algebraic numbers*(2nd, substantially revised and extended ed.), Springer-Verlag; PWN-Polish Scientific Publishers, ISBN 3-540-51250-0, Zbl 0717.11045 - Neukirch, Jürgen (1999).
*Algebraic Number Theory*.*Grundlehren der mathematischen Wissenschaften*.**322**. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. - Serre, Jean-Pierre (1979),
*Local Fields*, Graduate Texts in Mathematics,**67**, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, Zbl 0423.12016 - Weiss, Edwin (1976),
*Algebraic Number Theory*(2nd unaltered ed.), Chelsea Publishing, ISBN 0-8284-0293-0, Zbl 0348.12101