Today, we will prove the second inequality: the norm index of the ideles is at most the degree of the field extension. We will prove this using ideles (cf. the discussion of how ideles and ideals connect to each other), and some analysis.

1. A Big Theorem

We shall use one key fact from the theory of L-series. Namely, it is that:

Theorem 1 If ${k}$ is a number field, we have

$\displaystyle \sum_{\mathfrak{p}} \mathbf{N} \mathfrak{p}^{-s} \sim \log \frac{1}{s-1} (*)$

as ${s \rightarrow 1^+}$. Here ${\mathfrak{p}}$ ranges over the primes of ${k}$. The notation ${\sim}$ means that the two differ by a bounded quantity as ${s \rightarrow 1^+}$.

This gives a qualititative expression for what the distribution of primes must kinda look like—with the aid of some Tauberian theorems, one can deduce that the number of primes of norm at most ${N}$ is asymptotically ${N/\log N}$ for ${N \rightarrow \infty}$, i.e. an analog of the standard prime number theorem. In number fields. We actually need a slight refinement thereof.

Theorem 2 More generally, if ${\chi}$ is a character of the group ${I(\mathfrak{c})/P_{\mathfrak{c}}}$, we have$\displaystyle \sum_{ \mathfrak{p} \not\mid \mathfrak{c}} \chi(\mathfrak{p}) \mathbf{N} \mathfrak{p}^{-s} \sim \log \frac{1}{s-1}$

if ${\chi \equiv 1}$, and otherwise it tends either to a finite limit or ${-\infty}$.

Instead of just stating this as a random, isolated fact, I’d like to give some sort of context.  Recall that the Riemann-zeta function was defined as ${\zeta(s)=\sum_n n^{-s}}$. There is a generalization of this to number fields, called the Dedekind zeta function. The Dedekind-zeta function is not defined by summing over ${\sum |N(\alpha)|}$ for ${\alpha}$ in the ring of integers (minus 0). Why not? Because the ring of integers is not a unique factorization domain in general, and therefore we don’t get a nice product formula. (more…)