Different ideal

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

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

tr(xy)

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). Defining a fractional ideal I of K as the set of x ∈ K such that tr(xy) is an integer for all y in O_K, then I contains O_K. By definition, the different ideal δ_K is I⁻¹, an ideal of O_K.

The ideal norm of δ_K is the ideal of Z generated by the discriminant D_K of K.

The different is also defined for an 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. In a tower of fields L/K/F the relative differents are related by δ_L/F = δ_L/K δ_K/F.

The relative different equals the annihilator of the relative differential module $\Omega^1_{O_L/O_K}$.

Ramification

The relative different encodes the ramification data of the field extension L/K. A prime ideal p of K ramifies in L if and only if it divides the relative discriminant Δ_L/K. 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. 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. In the case when P is wildly ramified the differential exponent lies in the range e to e+ν_P(e)-1 and can be computed from the orders of the higher ramification groups for Galois extensions:

$\sum_{i=0}^\infty(|G_i|-1)$.

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

References

• Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-64767-6, MR1290116 . Translated from the original French by John Meldrum
• 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
• Frohlich, Albrecht; Taylor, Martin (1991), Algebraic number theory, Cambridge Studies in Advanced Mathematics, 27, Cambridge University Press, ISBN 0-521-36664-X 
• Weiss, Edwin (1976), Algebraic number theory, Chelsea Publishing, ISBN 0-8284-0293-0 .