Different ideal

In algebraic number theory, the different ideal is an invariant attached to an extension of algebraic number fields.

Let L/K be such an extension, with rings of integers O_L and O_K respectively. The relative trace defines a bilinear form on L by

$(x,y)\mapsto \mathrm {tr} _{L/K}(x\cdot y)\,$

in which the dual of O_L is a fractional ideal of L containing O_L. The (relative) different δ_L/K is the inverse of this fractional ideal: it is an ideal of O_L.

The relative norm of the relative different is 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.

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.

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).

References

• Weiss, Edwin (1976), Algebraic number theory, Chelsea Publishing, ISBN 0-8284-0293-0.
• Frohlich, Albrecht & Martin Taylor (1991), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 0-521-36664-X