Codifferent of a number field

The codifferent of a number field $K$ is

\mathcal O_K^\vee=\{a\in K:\operatorname{Tr}_{K/\mathbb Q}(ab)\in\mathbb Z\text{ for all }b\in\mathcal O_K\}.

It is the trace-dual lattice of $\mathcal O_K$ , also denoted $\mathfrak D_K^{-1}$ .

Trace-dual basis

For an integral basis $b_1,\ldots,b_m$ , the nondegenerate trace pairing gives a unique rational basis $b_1^\vee,\ldots,b_m^\vee$ with $\operatorname{Tr}(b_i^\vee b_j)=\delta_{ij}$ . The codifferent is their integer span. This is the basis realization of duality through the trace pairing.

Clearing denominators

Integrality of traces gives $\mathcal O_K\subseteq\mathcal O_K^\vee$ . Since its basis lies in $K$ , some integer $N\geq1$ satisfies $N\mathcal O_K^\vee\subseteq\mathcal O_K$ . These inclusions allow equivalent ergodicity tests with integral coefficients, though the complete character lattice is still the codifferent.

Gaussian integers

For $K=\mathbb Q(i)$ , testing against $1,i$ gives $\mathcal O_K^\vee=\tfrac12\mathbb Z[i]$ . The codifferent need not equal the ring of integers.

References

Keith Conrad, The Different Ideal. Definition 3.2, Theorem 3.4, and §4.