Places of an algebraic number field

Question

I have been doing my best to learn the rudiments of class field theory via the formulation in terms of ideals.

The $\textbf{ray class group}$ of an algebraic number field $K$ with respect to $\mathfrak{m}$ is defined as:

$$ \textrm{Cl}_{K}^{\mathfrak{m}}:=J^{\mathfrak{m}}/P^{\mathfrak{m}}.$$

I was initially under the erroneous impression that the ray class group is defined in terms of any ideal $\mathfrak{m} \subset \mathcal{O}_K$, so that $J^{\mathfrak{m}}$ is the group of ideals in $\mathcal{O}_K$ coprime to $\mathfrak{m}$ and $P^{\mathfrak{m}}$ is the normal subgroup of prime ideals congruent to $1\ (\textrm{mod}\ \mathfrak{m})$.

$\textrm{Cl}_{K}^{\mathfrak{m}}$ would then be a generalisation of the ideal class group - coinciding with the latter when $\mathfrak{m}=1$.

After further elucidation, it turns out that the ray class group is actually defined in terms of something called a $\textbf{modulus}$ $\mathfrak{m}$ of $K$, defined as: $$ \mathfrak{m} := \prod_{\mathfrak{p}}\mathfrak{p}^{\nu(\mathfrak{p})} \quad\textrm{where}\ \nu(\mathfrak{p}) \neq 0\ \textrm{for finitely many}\ \mathfrak{p}. \tag{$\dagger$} $$ When I first saw ($\dagger$), I thought to myself: Fair enough; the product is taken over all prime ideals $\mathfrak{p} \subset \mathcal{O}_K$, so $\mathfrak{m}$ is just some ideal in $\mathcal{O}_K$, and "modulus" is just fancy nomenclature.

But upon closer inspection, it turns out that the modulus is not defined as a product of prime ideals, but rather in terms of something called the $\textbf{places}$ of $K$.

Thus I set out to find out exactly what is meant by the "places" of a number field $K$. And in my quest to understand the notion of places, I have managed to find places in the literature where places are mentioned, but no comprehensive treatment or even a definition. As for the online resources, I have found them wanting in intelligibility and consistency.

Most resources I have consulted treat something called Archimedean places, which they relate to the real and complex embeddings of the number field in question.

Hence my questions - which may be somewhat soft - are the following:

1) Why is the ray class group defined in terms of these moduli, as opposed to ideals. $\textit{E.g.}$ if we defined it in terms of an ideal $\mathfrak{m}$, would it still be true that every ray class group had a corresponding ray class field, $\textit{i.e.}$ a normal extension $E$ of $K$, so that $\textrm{Cl}_{K}^{\mathfrak{m}} \cong \textrm{Gal}(E/K)$. That is to say: Would we still have the Takagi existence theorem?

2) Where can I find a really thorough treatment of the notion of a place of a number field?

3) As with most things in mathematics, I believe that it is most instructive to confront a few concrete cases before one gives an abstract definition. So can we provide concrete instances of "places" of $\mathbb{Q}$, of $\mathbb{Q}(i)$ or of $\mathbb{Q}(\sqrt{2})$?

Answer 1

I won’t help you with the ray-class group, but a place is either (a) a maximal ideal of the ring of integers of $K$ or (b) an equivalence class of archimedean metrics on the field.

For $\Bbb Q$, the places are the ordinary primes plus the “infinite” place of the standard absolute value you used in Calculus.

For $\Bbb Q(i)$, the places are the single prime $(1+i)$ dividing $(2)$; the primes $(a\pm bi)$ for $p=a^2+b^2$ a prime $\equiv1\pmod4$; and the ordinary primes $\equiv3\pmod4$. Again there’s only one infinite place, the familiar absolute value on $\Bbb C$.

For $\Bbb Q(\sqrt2\,)$, there again is only one ramified prime $(\sqrt2\,)$, and the other natural primes split or don’t according as $p\equiv\pm1\pmod8$ for the split ones; or $p\equiv\pm3\pmod8$ for the ones that remain prime. For instance, since $7$ is of form $a^2-2n^2$ for $a=3$, $b=1$, we may write $(7)=(3+\sqrt2\,)(3-\sqrt2\,)$. So there are two places of $K=\Bbb Q(\sqrt2\,)$ above $(7)$, while $(3)$ remains prime in $K$, and there’s just the single place $(3)$ above three. The archimedean primes are more fun. You can embed $K$ into $\Bbb R$ in two ways, by sending a particular $\lambda$ for which $\lambda^2=2$ to the positive or the negative square root of $2\in\Bbb R$. So that gives you two inequivalent archimedean metrics on $\Bbb Q(\sqrt2\,)$.

In general, if $[K:\Bbb Q]=n$, there will be at most $n$ places of $K$ above any place of $\Bbb Q$. Hope this helps.

Answer 2

All the technical semantics are due to the infinite primes, or infinite places, or achimedean primes (or whatever) when defining an infinite prime of an arbitrary number field K.

For any embedding $\sigma$ of K into an algebraic closure, the infinite prime $P_{\sigma}$ is just a formal symbol attached to $\sigma$. A problem arises when considering an extension $L/K$ s.t. $\sigma$ embeds K into $\mathbf R$, but extends to 2 conjugate complex embeddings $\tau, \tau '$ of L into $\mathbf C$. In this case one defines the infinite prime $P_{\tau}$ over $P_{\sigma}$ as being the formal symbol $P_{\tau}=P_{\tau'}$, and one says $P_{\tau}$ is ramified over $P_{\sigma}$. The general statements about ideals are thus extended coherently if we look at the completions ($\mathbf R$ and $\mathbf C$ here), but incoherently if we consider splitting phenomena (e.g. a finite prime ideal P of K which splits as Q.Q' in L, in which case Q is unramified above P in the classical parlance).

A modulus of K is by definition a formal product $M=M_0.M_{\infty}$, where $M_0$ is an ordinary ideal and $M_{\infty}$ is a formal product of real infinite primes of K (all the infinite primes are raised to the first power here.) Then the ray class-field $K_{M}$ is the maximal abelian extension of K which is unramified outside $M$, and Takagi's existence theorem does need modulii. An excellent account of all this, with many examples, is §7 of D. Garbanati, "CFT summarized", Rocky Mountain J. of Math., 11,2(1981), 195-225.