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…)