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

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

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.

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 / 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]

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 = 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]

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

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

1. Dedekind 1882
2. Bourbaki 1994, p. 102
3. Serre 1979 , p. 50
4. Fröhlich & Taylor 1991 , p. 125
5. Neukirch 1999 , p. 195
6. Narkiewicz 1990 , p. 160
7. Hecke 1981 , p. 116
8. Hecke 1981 , p. 121
9. Neukirch 1999 , pp. 197–198
10. Neukirch 1999 , p. 201
11. Fröhlich & Taylor 1991 , p. 126
12. Serre 1979 , p. 59
13. Hecke 1981 , pp. 234–236
14. Narkiewicz 1990 , p. 304
15. Narkiewicz 1990 , p. 401
16. Neukirch 1999 , pp. 199
17. Narkiewicz 1990 , p. 166
18. Weiss 1976, p. 114
19. Narkiewicz 1990 , pp. 194, 270
20. 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, 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