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. will be a number field, and an absolute value (which, by abuse of terminology, we will use interchangeable with “valuation” and “place”) extending one of the absolute values on (which are always normalized in the standard way); we will write for the output at .

Suppose is a valuation of ; we write if extends . Recall the following important formula from the theory of absolute values an extension fields:

Write ; these are the **local degrees**. From elementary algebraic number theory, we have This is essentially a version of the formula.

The above formalism allows us to deduce an important global relation between the absolute values for .

Theorem 1 (Product formula)If ,

The proof of this theorem starts with the case , in which case it is an immediate consequence of unique factorization. For instance, one can argue as follows. (more…)