Number-field torus - Knowlpedia

Core idea

For a number field $K$ of degree $m$ , the number-field torus is the compact additive group

\mathbb T_K=K_\infty/\iota(\mathcal O_K).

Its $d$ -fold power is $X=K_\infty^d/\iota(\mathcal O_K)^d$ , a real torus of dimension $md$ , endowed with normalized Haar measure and, when needed, the completed Borel sigma-algebra.

Why all Archimedean embeddings are needed

The integral lattice has full real rank in $K_\infty$ . An integral basis identifies this quotient as a topological group with $(\mathbb R/\mathbb Z)^m$ . These are quotients of additive groups; the integral lattice is not an ideal in the Archimedean algebra, so no quotient-ring structure is being asserted.

Torsion points

Regard the additive torus as a $\mathbb Z$ -module. Its torsion elements, called torsion points, are exactly the points killed by a nonzero integer.

The smaller quotient $K/\mathcal O_K\cong(\mathbb Q/\mathbb Z)^m$ is exactly the torsion subgroup inside $\mathbb T_K$ : $N[x]=0$ implies $Nx\in\mathcal O_K$ , hence $x\in K$ , and the converse follows by clearing denominators. Thus $K/\mathcal O_K$ is not the full torus. An increment from $K^d$ produces a periodic translation.

References

J. S. Milne, Algebraic Number Theory, version 3.08, 2020. Chapter 4, lattice embedding; the quotient and torsion assertions follow from the displayed integral-basis calculation.