It seems that almost every inquiry into an area of number theory begins with the fact that an integer can be uniquely factored into powers of primes, upto the order of the primes. In more general settings, one defines an element to be prime if $p \mid ab$ implies that $p \mid a$ or $p \mid b$. Consider the ring of Gaussian integers: \[ \mathbb{Z}[i] = \left\{a + bi \mid a,b \in \mathbb{Z} \right\}. \] What are the prime elements in this ring? A natural enough question that one can answer by working through the properties of the norm endowed on the ring $N(z) = z \cdot \bar{z}$, where $\bar{z}$ is the complex conjugate of $z$.
Consider the following diagram of ring / field extensions: \[ \require{AMScd} \begin{CD} \langle 1-i \rangle @>>> \mathcal{O}_K = \mathbb{Z}[i] @>>> K = \mathbb{Q}(i) \\ @AAA @AAA @AAA \\ \langle 2 \rangle @>{\phantom{\text{extension}}}>> \mathbb{Z} @>{\phantom{\text{extension}}}>> \mathbb{Q} \\ \end{CD} \] Note that $2$ lies in the ideal $\langle 1-i \rangle$ because $(1+i)(1-i) = 2$. One says that $2$ is a split prime, in the extension, since it factors. It turns out that there can only be finitely many such primes. There are three types of primes split , ramified , and inert . They are interesting enough, in isolation, but really one wants to use the behavior to study the ring, at large. One can then ask: does unique factorization hold in every number field $\mathcal{O}_K$?
The answer turns out to be no. Not every number field is a unique factorization domain(UFD). However, they are Dedekind domains . These rings have suitably nice properties to recover a version of unique factorization. One looks at prime ideals instead of prime elements.
Theorem: Every ideal, in a Dedekind domain, uniquely factors into a product of prime ideals, up to reordering of the factors.
This is but the beginning of a very interesting story. In some ways, the "algebraic", in algebraic number theory, is a little bit of a misnomer. There are analytic notions that come into play, while proving certain theorems or trying to understand the structure of a number field. Two examples are the Dedekind Unit theorem and Dedekind Zeta functions. The Dedekind Unit theorem requires analysis, in its proof, but is ultimately a statement about the units in a number ring.
There are a whole slew of ideas, some of which are pretty deep, that lead off into the realm of Class Field Theory. The class number tells one how far a number field is from being a PID, at least in some sense. The class number formula relates values of the Dedekind zeta function to invariants such as the regulator the discriminant , the number of the roots of unity in the number field, and the real and complex embeddings.
The Dedekind Zeta Function: We take a small look regarding the intersection between trying to uncover the algebraic structure of a number ring, and analytic techniques. As noted in the analytic number theory section, the Riemann-Zeta function is defined by: \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \text{ where } \Re(s) > 1. \] There is an analog to this idea, taken from an algebraic point-of-view. It was Hecke who first defined the Dedekind-Zeta function, as follows: \[ \zeta_K(s) = \sum_{I} \frac{1}{N(I)^s} \] where the sum is taken over the integral ideals in $\mathcal{O}_K$. One way to define the norm is by taking $N(I)$ to be the size of the quotient ring $\mathcal{O}_K/I$. Landau was able to prove the following analog to the Prime Number Theorem:
Prime Ideal Theorem: Let $K$ be an algebraic number field. The number of prime ideals $\mathfrak{p}$, with $N(\mathfrak{p}) \leq x$ is asymptotic to $x/\log x$ as $x \rightarrow \infty$.
Q: If the Riemann-Zeta function and the Dedekind-Zeta function have the following traits in common:
Euler Product: The intuition behind why there is an Euler product is via the generalization of the Fundamental Theorem of Arithmetic. Again, we are using the fact that ideals factors uniquely in a Dedekind domain, along with the multiplicativity of the norm. This is the same idea used when studying Dirichlet series. We will avoid issues of convergence, to insure that we don't get bogged down in the technicalities and miss the story. Just as with the Riemann-Zeta function, the series representation \[ \zeta_K(s) = \sum_{I} \frac{1}{N(I)^s} \] yields the product \[ \prod_{\mathfrak{p}} \left(1 + \frac{1}{N(\mathfrak{p}^s)} + \frac{1}{N(\mathfrak{p}^{2s})} + cdots \right) \] where the product is taken over the prime ideals $\mathfrak{p}$ in the ring of integers $\mathcal{O}_K$. The idea is that when the product is multiplied out term-by-term, each ideal $I$ will appear exactly once. Now notice that, for each prime ideal $\mathfrak{p}$, the factor is a geometric series. This, at least intuitively, justifies our writing \[ \zeta_K(s) = \prod_{\mathfrak{p}} \left(1 - \frac{1}{N(\mathfrak{p})^s} \right)^{-1} \] for $\Re(s) > 1$. The following theorem gives us an answer to (2), as well.
Theorem: Let $K$ be a number field. Then $\zeta_k$ is a meromorphic function on the half-plane $\Re(s) > 1 - \frac{1}{[K:\mathbb{Q}]}$, analytic everywhere except for the simple pole at $s=1$. Further, \[ \zeta_K(s) = \sum_I \frac{1}{N(I)^s} = \prod_{\mathfrak{p}} \left(1 - \frac{1}{N(\mathfrak{p})^s}\right)^{-1} \] with absolute convergence for $\Re(s) > 1$.
The second question gives us a connection between the algebraic structure of $\mathcal{O}_K$, the analytic properties of $\zeta_K$, and the class number. It turns out that one can write \[ \zeta_K(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s} \] where the $a_n$ are the number of ideals with norm $n$. One can proceed from here to get the class number formula. See Marcus Chapter 7. In the exercises in Montgomery and Vaugh, the ideas above are formulated for what happens in for quadratic number fields.
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