This post won’t be as cool as the title sounds. But I will prove something neat, and it will lead to neater things as time goes on (assuming I keep posting on this topic).

We will now return to algebraic number theory, following Lang’s textbook, and study the distribution of points in parallelotopes.

The setup is as follows. ${K}$ will be a number field, and ${v}$ an absolute value (which, by abuse of terminology, we will use interchangeable with “valuation” and “place”) extending one of the absolute values on ${\mathbb{Q}}$ (which are always normalized in the standard way); we will write ${|x|_v}$ for the output at ${x \in K}$.

Suppose ${v_0}$ is a valuation of ${\mathbb{Q}}$; we write ${v | v_0}$ if ${v}$ extends ${v_0}$. Recall the following important formula from the theory of absolute values an extension fields:

$\displaystyle |N^K_{\mathbb{Q}}(x)|_{v_0} = \prod_{v | v_0} |x|_v^{ [K_v: \mathbb{Q}_{v_0} ] }.$

Write ${N_v := [K_v: \mathbb{Q}_{v_0} ] }$; these are the local degrees. From elementary algebraic number theory, we have ${\sum_{v | v_0} N_v = N := [K:\mathbb{Q}].}$ This is essentially a version of the ${\sum ef = N}$ formula.

The above formalism allows us to deduce an important global relation between the absolute values ${|x|_v}$ for ${x \in K}$.

Theorem 1 (Product formula) If ${x \neq 0}$,$\displaystyle \prod_v |x|_v^{N_v} = 1.$

The proof of this theorem starts with the case ${K=\mathbb{Q}}$, in which case it is an immediate consequence of unique factorization. For instance, one can argue as follows. (more…)