Mathematics Lecture Notes

TheoremExistence Theorem

For every modulus $\mathfrak{m}$ and subgroup $H \subseteq \text{Cl}_\mathfrak{m}(K)$ of finite index, there exists a unique abelian extension $L/K$ such that: $\Phi_{L/K}(\text{Cl}_\mathfrak{m}(K)) = H$

This extension is called the class field corresponding to $H$ . The degree $[L : K]$ equals the index $[\text{Cl}_\mathfrak{m}(K) : H]$ .

DefinitionHilbert Class Field

The Hilbert class field $H_K$ is the class field corresponding to $H = \{1\}$ in $\text{Cl}(K)$ (the ideal class group).

Properties:

$\text{Gal}(H_K/K) \cong \text{Cl}(K)$
$H_K/K$ is unramified at all primes (including infinite primes)
$\mathcal{O}_{H_K}$ has class number 1 (is a PID)
Every ideal of $K$ becomes principal in $H_K$

This is the maximal unramified abelian extension of $K$ .

ExampleHilbert Class Fields

$K = \mathbb{Q}(i)$ : $h_K = 1$ , so $H_K = K$ (no proper extension)
$K = \mathbb{Q}(\sqrt{-5})$ : $h_K = 2$ , and $H_K = \mathbb{Q}(\sqrt{-5}, i) = \mathbb{Q}(\sqrt{-1}, \sqrt{5})$
$K = \mathbb{Q}(\sqrt{-23})$ : $h_K = 3$ , and $H_K$ is a cyclic cubic extension

Computing $H_K$ explicitly is a major challenge in computational number theory.

TheoremChebotarev Density Theorem

Let $L/K$ be Galois. For any conjugacy class $C \subseteq \text{Gal}(L/K)$ , the set of primes $\mathfrak{p}$ with $\text{Frob}_\mathfrak{p} \in C$ has natural density: $\delta(\{\mathfrak{p} : \text{Frob}_\mathfrak{p} \in C\}) = \frac{|C|}{|\text{Gal}(L/K)|}$

For abelian extensions, this becomes Dirichlet's theorem on primes in arithmetic progressions. This powerful generalization determines splitting behavior statistically.

Remark

The Artin map connects splitting of primes to ideal classes. A prime $\mathfrak{p}$ splits completely in $L/K$ if and only if its class in $\text{Cl}_\mathfrak{m}(K)$ lies in the kernel of the Artin map, i.e., in the subgroup corresponding to $L/K$ .

This criterion enables explicit determination of which primes split, providing computational methods for class field constructions.

TheoremNorm Limitation Theorem

For an abelian extension $L/K$ , the conductor $\mathfrak{m}$ satisfies: $N_{L/K}(I_\mathfrak{m}(L)) = P_\mathfrak{m,1}(K)$

where $P_\mathfrak{m,1}$ consists of principal ideals $(\alpha)$ with $\alpha \equiv 1 \pmod{\mathfrak{m}}$ and totally positive at real places in $\mathfrak{m}$ .

This norm limitation characterizes which ideals are norms from the extension, crucial for existence and uniqueness theorems.

Math Notes

Class Field Theory - Key Properties