I am reading Cox's Primes of the form $x^2+ny^2$. There, he's given the following definition:
Given a number field $K$, a modulus in $K$ is a formal product $$\mathfrak{m}=\prod_{\mathfrak{p}}\mathfrak{p}^{n_{\mathfrak{p}}}$$ over all primes $\mathfrak{p}$, finite or infinite, where the exponents must satisfy
- $n_{\mathfrak{p}} \ge0$, and at most finitely many are non-zero.
- $n_{\mathfrak{p}}=0$ whenever $\mathfrak{p}$ is an infinite complex prime.
- $n_{\mathfrak{p}}\le1$ whenever $\mathfrak{p}$ is a infinite real prime.
A modulus may thus be written as written as $\mathfrak{m}_{0}\mathfrak{m}_{\infty}$ where $\mathfrak{m}_{0}$ is an $\mathcal{O}_K$-ideal and $\mathfrak{m}_{\infty}$ is a product of distinct real infinite primes of $K$.
I am confused as to what it means to take products of infinite primes? Essentially, one is just taking the product of Archimedean places/absolute values, and taking the product of absolute values seems absurd to me. If we consider them as embeddings, I'd imagine the "product" would be "composition"? But then how does one multiply functions with ideals? Or maybe I am just understanding infinite primes all wrong again :(
For context, the book defines infinite primes as mentioned here.
It is just a bookkeeping device: you want to associate to each nonzero prime ideal $\mathfrak p$ a "multiplicity" $n_\mathfrak p$, which is a nonnegative integer and $n_\mathfrak p = 0$ for all but finitely many $\mathfrak p$. You also want to keep track of real primes (real embeddings) and you don't care about complex primes.
Thanks to unique factorization of ideals, all the data at the nonzero prime ideals can be encoded by the product of ideals $\prod_{\mathfrak p} \mathfrak p^{n_\mathfrak p}$. But if we only knew about prime ideals and had no concept of multiplication of ideals, then we would just speak about a function from prime (both finite and real/complex) to nonnegative integers ($\mathfrak p \mapsto n_\mathfrak p$) that has nonzero values only finitely many times. Since we do know about multiplication of ordinary ideals, we can make sense of that product over finite primes, and to allow this approach to keep track of real primes we have to allow formal multiplication of them in the notation.