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 OK is the ring of integers of K, and tr denotes the field trace from K to the rational number field Q, then
- [math]\displaystyle{ x \mapsto \mathrm{tr}~x^2 }[/math]
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 δK is the inverse fractional ideal I−1: it is an ideal of OK.
The ideal norm of δK is equal to the ideal of Z generated by the field discriminant DK 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
- [math]\displaystyle{ \delta(\alpha) = \prod \left({\alpha - \alpha^{(i)}}\right) \ }[/math]
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 OK.[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 δL / 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 ΔL / K.[10] In a tower of fields L / K / F the relative differents are related by δL / F = δL / KδK / F.[5][11]
The relative different equals the annihilator of the relative Kähler differential module [math]\displaystyle{ \Omega^1_{O_L/O_K} }[/math]:[10][12]
[math]\displaystyle{ \delta_{L/K} = \{ x \in O_L : x \mathrm{d} y = 0 \text{ for all } y \in O_L \} . }[/math]
The ideal class of the relative different δL / K is always a square in the class group of OL, 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 OK:[14] indeed, it is the square of the Steinitz class for OL as a OK-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 ΔL / K. More precisely, if
- p = P1e(1) ... Pke(k)
is the factorisation of p into prime ideals of L then Pi divides the relative different δL / K if and only if Pi 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 + eνP(e) − 1.[16][18][19] The differential exponent can be computed from the orders of the higher ramification groups for Galois extensions:[20] [math]\displaystyle{ \sum_{i=0}^\infty (|G_i|-1). }[/math]
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
- ↑ 5.0 5.1 Neukirch 1999, p. 195
- ↑ 6.0 6.1 Narkiewicz 1990, p. 160
- ↑ Hecke 1981, p. 116
- ↑ Hecke 1981, p. 121
- ↑ Neukirch 1999, pp. 197–198
- ↑ 10.0 10.1 Neukirch 1999, p. 201
- ↑ 11.0 11.1 Fröhlich & Taylor 1991, p. 126
- ↑ Serre 1979, p. 59
- ↑ Hecke 1981, pp. 234–236
- ↑ Narkiewicz 1990, p. 304
- ↑ Narkiewicz 1990, p. 401
- ↑ 16.0 16.1 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. Berlin: Springer-Verlag. ISBN 978-3-540-64767-6.
- Dedekind, Richard (1882), "Über die Discriminanten endlicher Körper", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 29 (2): 1–56, http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=39145. 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
- Hecke, Erich (1981), Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, 77, New York–Heidelberg–Berlin: Springer-Verlag, ISBN 3-540-90595-2
- 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, https://archive.org/details/elementaryanalyt0000nark
- Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8.
- Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, ISBN 0-387-90424-7
- Weiss, Edwin (1976), Algebraic Number Theory (2nd unaltered ed.), Chelsea Publishing, ISBN 0-8284-0293-0, https://archive.org/details/algebraicnumbert00weis_0
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