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

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

is an

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

\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

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 $\Omega _{O_{L}/O_{K}}^{1}$ :^[10]^[12]

$\delta _{L/K}=\{x\in O_{L}:x\mathrm {d} y=0{\text{ for all }}y\in O_{L}\}.$

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.

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]