Mathematics Lecture Notes

TheoremPrincipal Ideal Theorem

Let $K$ be a number field with class number $h_K$ . For any ideal $\mathfrak{a}$ in $\mathcal{O}_K$ , the ideal $\mathfrak{a}^{h_K}$ is principal.

Moreover, in the Hilbert class field $H_K$ , the ideal $\mathfrak{a}\mathcal{O}_{H_K}$ becomes principal. That is, every ideal of $K$ becomes principal when lifted to $H_K$ .

This explains why $\mathcal{O}_{H_K}$ always has class number 1.

ExampleSolving $x^3 = y^2 + 2$ via Class Groups

Consider the equation $x^3 = y^2 + 2$ over integers. Working in $K = \mathbb{Q}(\sqrt{-2})$ : $(y + \sqrt{-2})(y - \sqrt{-2}) = x^3$

Since $h_K = 1$ (class number is 1), $\mathcal{O}_K = \mathbb{Z}[\sqrt{-2}]$ is a UFD. If $\gcd(y + \sqrt{-2}, y - \sqrt{-2}) = 1$ , each factor must be a perfect cube: $y + \sqrt{-2} = (a + b\sqrt{-2})^3$

Expanding and solving gives $(x, y) = (3, \pm 5)$ as the only integer solutions.

TheoremEmbedding Problem

Let $K$ be a number field and $G$ a finite group. An extension $L/K$ with $\text{Gal}(L/K) \cong G$ exists unramified outside a finite set $S$ of primes if and only if a certain cohomological condition on $\text{Cl}(K)$ is satisfied.

The class group acts as an obstruction: when $h_K = 1$ , many embedding problems simplify.

ExamplePell's Equation and Class Numbers

The continued fraction expansion of $\sqrt{d}$ relates to units and class numbers of $\mathbb{Q}(\sqrt{d})$ .

For $d = 13$ : the fundamental solution to $x^2 - 13y^2 = 1$ is $(649, 180)$ , giving fundamental unit $\epsilon = 649 + 180\sqrt{13}$ with large regulator.

The class number $h_K = 1$ , and the large regulator compensates in the class number formula.

TheoremApplications to Cryptography

Modern lattice-based cryptography uses class groups:

Buchmann-Williams key exchange: Uses infrastructure of the class group
CSIDH protocol: Isogeny-based scheme using supersingular elliptic curve class groups
Ideal lattices: NTRU and Ring-LWE use ideal structures in cyclotomic fields

Security relies on computational hardness of:

Computing class group structure
Solving discrete log in $\text{Cl}(K)$
Finding short vectors in ideal lattices

ExampleFactorization Algorithms

The quadratic sieve and number field sieve use class groups implicitly:

To factor $N$ , select smooth elements $\alpha \in \mathcal{O}_K$ with $N(\alpha)$ smooth (factors into small primes). Relations among principal ideals $(\alpha)$ in $\text{Cl}(K)$ provide dependencies yielding factors of $N$ .

This approach factored RSA-768 (232-digit number) in 2009.

TheoremBSD Conjecture Connection

For elliptic curves $E$ over number fields $K$ , the Birch and Swinnerton-Dyer conjecture predicts: $\lim_{s \to 1} \frac{L(E/K, s)}{(s-1)^r} = \frac{\# Sha(E/K) \cdot \Omega_E \cdot R_E \cdot \prod_p c_p}{(\# E(K)_{\text{tors}})^2}$

The mysterious Shafarevich-Tate group $Sha(E/K)$ is conjectured finite, analogous to the class group. Like $\text{Cl}(K)$ for number fields, $Sha(E/K)$ measures obstruction to local-global principles for elliptic curves.

Remark

Class groups appear throughout modern number theory:

Iwasawa theory: Studies growth of class numbers in $\mathbb{Z}_p$ -extensions
Euler systems: Constructs cohomology classes related to class groups
Modularity: Class numbers appear in coefficients of modular forms

These connections make class group computations central to computational number theory.

Math Notes

The Class Group - Applications