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.

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

x

↦

t

r

x

2

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

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 or Dedekind's complementary module 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): we may write

δ

(

α

)

=

∏

(

α

−

α

(

i

)

)

{\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. The different ideal is generated by the differents of all integers α in OK. This is Dedekind's original definition.

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. In a tower of fields L / K / F the relative differents are related by δL / F = δL / KδK / F.

The relative different equals the annihilator of the relative Kähler differential module

Ω

O

L

/

O

K

1

{\displaystyle \Omega _{O_{L}/O_{K}}^{1}}

:

δ

L

/

K

=

{

x

∈

O

L

:

x

d

y

=

0

for all

y

∈

O

L

}

.

{\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 δL / K is always a square in the class group of OL, the ring of integers of L. Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of OK: indeed, it is the square of the Steinitz class for OL as a OK-module.