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
[edit]If OK 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 OK. 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 OK, then I is a fractional ideal of K containing OK. By definition, the different ideal
The ideal norm of
The different of an element
where the
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
[edit]The relative different
The relative different equals the annihilator of the relative Kähler differential module :[10][12]
The ideal class of the relative different
Ramification
[edit]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
- p = P1e(1) ... Pke(k)
is the factorisation of p into prime ideals of L then Pi divides the relative different
Local computation
[edit]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
Notes
[edit]- ^ Dedekind 1882
- ^ Bourbaki 1994, p. 102
- ^ Serre 1979, p. 50
- ^ Fröhlich & Taylor 1991, p. 125
- ^ a b Neukirch 1999, p. 195
- ^ a b Narkiewicz 1990, p. 160
- ^ Hecke 1981, p. 116
- ^ Hecke 1981, p. 121
- ^ Neukirch 1999, pp. 197–198
- ^ a b Neukirch 1999, p. 201
- ^ a b Fröhlich & Taylor 1991, p. 126
- ^ Serre 1979, p. 59
- ^ Hecke 1981, pp. 234–236
- ^ Narkiewicz 1990, p. 304
- ^ Narkiewicz 1990, p. 401
- ^ a b Neukirch 1999, pp. 199
- ^ Narkiewicz 1990, p. 166
- ^ Weiss 1976, p. 114
- ^ Narkiewicz 1990, pp. 194, 270
- ^ Weiss 1976, p. 115
References
[edit]- 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