Archimedean trace pairing

On $K_\infty=\mathbb R^{r_1}\times\mathbb C^{r_2}$ , define

\operatorname{Tr}_\infty(x_1,\ldots,x_{r_1},z_1,\ldots,z_{r_2}) =\sum_jx_j+2\sum_j\operatorname{Re}z_j.

The Archimedean trace pairing is the nondegenerate real bilinear form $(a,x)\mapsto\operatorname{Tr}_\infty(ax)$ . On embedded field elements the trace agrees with $\operatorname{Tr}_{K/\mathbb Q}$ .

It is not a Hermitian inner product

The complex-coordinate product is $az$ , without conjugation. In particular, $2\operatorname{Re}(az)$ can be negative for $a=z$ . Nondegeneracy, rather than positive definiteness, is what identifies the dual lattice used in Fourier characters.

For vectors, write $a\cdot x=\sum_{j=1}^d a_jx_j$ and pair by $\operatorname{Tr}_\infty(a\cdot x)$ .

References

J. S. Milne, Algebraic Number Theory, version 3.08, 2020. Chapter 2, “Review of norms and traces.”