Different ideal

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

${\displaystyle x\mapsto \mathrm {tr} ~x^{2}}$

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

${\displaystyle \delta (\alpha )=\prod \left({\alpha -\alpha ^{(i)}}\right)\ }$

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.

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 / Kδ_K / F.[5][11]

The relative different equals the annihilator of the relative Kähler differential module ${\displaystyle \Omega _{O_{L}/O_{K}}^{1}}$:[10][12]

${\displaystyle \delta _{L/K}=\{x\in O_{L}:x\mathrm {d} y=0{\text{ for all }}y\in O_{L}\}.}$

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]

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 = P₁^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 + eν_P(e) − 1.[16][18][19] The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[20]

$${\displaystyle \sum _{i=0}^{\infty }(|G_{i}|-1).}$$

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).

• 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, vol. 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, vol. 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). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 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, vol. 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