Algebraic Number Theory — Mathematics Knowledge System

Overview

Algebraic number theory studies the arithmetic of algebraic number fields — finite extensions of the rational numbers. It generalises classical integer arithmetic by replacing $\mathbb{Z}$ with rings of algebraic integers, revealing deep structure through ideals, class groups, and unit groups.

Intuition

In $\mathbb{Z}[\sqrt{-5}]$ the number 6 factors as both $2 \cdot 3$ and $(1+\sqrt{-5})(1-\sqrt{-5})$, breaking unique factorisation. Passing from elements to ideals restores it: the ideal $(6)$ factors uniquely as $\mathfrak{p}_2 \mathfrak{p}_3 \overline{\mathfrak{p}_3}$. The class group measures how far the ring is from having unique factorisation.

Formal Definition

Definition

Let $K$ be a number field of degree $n$ over $\mathbb{Q}$. The ring of integers $\mathcal{O}_K$ is the integral closure of $\mathbb{Z}$ in $K$.

\mathcal{O}_K = \{ \alpha \in K : f(\alpha) = 0 \text{ for some monic } f \in \mathbb{Z}[x] \}

Ring of integers of a number field

ring-of-integers

\text{Cl}(K) = \mathcal{I}_K / \mathcal{P}_K

Ideal class group: fractional ideals modulo principal ideals

class-group

h_K = |\text{Cl}(K)|

Class number; h_K = 1 iff unique factorisation holds in O_K

class-number

Notation

Notation	Meaning
$\mathcal{O}_K$	Ring of integers of the number field K
$\text{Cl}(K)$	Ideal class group of K
$h_K$	Class number of K
$N(\mathfrak{a})$	Norm of ideal a, equal to \|O_K/a\|
$\text{disc}(K)$	Discriminant of the number field

Theorems

Theorem 1: Dedekind's Theorem (unique factorisation of ideals)

Every non-zero ideal of \mathcal{O}_K factors uniquely as a product of prime ideals.

Theorem 2: Minkowski's Bound

Every ideal class contains an integral ideal of norm at most M_K = \frac{n!}{n^n}\left(\frac{4}{\pi}\right)^{r_2}\sqrt{|\text{disc}(K)|}.

Theorem 3: Dirichlet's Unit Theorem

\mathcal{O}_K^\times \cong \mu_K \times \mathbb{Z}^{r_1 + r_2 - 1}, where r_1 is the number of real embeddings, r_2 the number of complex conjugate pairs, and \mu_K the group of roots of unity in K.

Worked Examples

Since $-5 \equiv 3 \pmod{4}$, the ring of integers is $\mathbb{Z}[\sqrt{-5}]$ (not the larger $\mathbb{Z}[(1+\sqrt{-5})/2]$ which applies only when $d \equiv 1 \pmod{4}$).
$\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$
Minkowski's bound: $M_K = \frac{2}{\pi}\sqrt{20} \approx 2.85$, so we need only check primes 2.
$M_K = \frac{2}{\pi}\sqrt{|{-20}|} \approx 2.85$
The ideal $(2)$ splits: $(2) = (2, 1+\sqrt{-5})^2$. The prime ideal $\mathfrak{p} = (2, 1+\sqrt{-5})$ satisfies $\mathfrak{p}^2 = (2)$ but $\mathfrak{p}$ is not principal (there is no element of norm 2 in $\mathcal{O}_K$). Thus $h_K = 2$.

Answer: $\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$ and $h_K = 2$.

Practice Problems

Difficulty 7/10

Find the ring of integers of $\mathbb{Q}(\sqrt{3})$ and determine whether it has unique factorisation (i.e., compute $h_K$).

Difficulty 9/10

Prove that every non-zero prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ lies over a unique rational prime $p$, i.e., $\mathfrak{p} \cap \mathbb{Z} = (p)$.

Difficulty 8/10

Use Dirichlet's Unit Theorem to determine the structure of $\mathcal{O}_K^\times$ for $K = \mathbb{Q}(\sqrt{2})$.

Common Mistakes

Common Mistake

Assuming unique factorisation of elements holds in all rings of integers

Unique factorisation of elements fails whenever $h_K > 1$; it is ideals that factor uniquely.

Common Mistake

Confusing the ring of integers $\mathcal{O}_K$ with $\mathbb{Z}[\alpha]$ for a chosen generator $\alpha$

$\mathbb{Z}[\alpha] \subseteq \mathcal{O}_K$ but equality holds only when $\{1, \alpha, \ldots, \alpha^{n-1}\}$ is an integral basis.

Historical Background

Kummer introduced ideal numbers in the 1840s to repair a flawed proof of Fermat's Last Theorem, observing that unique factorisation fails in cyclotomic rings. Dedekind recast Kummer's ideas into the language of ideals in 1871, and Hilbert systematised the theory in his Zahlbericht of 1897. The 20th century brought class field theory (Takagi, Artin) and ultimately the Langlands programme.

1847
Kummer introduces ideal numbers for cyclotomic fields
Ernst Kummer
1871
Dedekind defines ideals and proves unique factorisation of ideals
Richard Dedekind
1897
Hilbert publishes the Zahlbericht
David Hilbert
1920s
Takagi and Artin develop class field theory
Teiji Takagi, Emil Artin

Summary

Algebraic number theory studies arithmetic in rings of integers $\mathcal{O}_K$ of number fields $K/\mathbb{Q}$.
Unique factorisation of elements may fail, but Dedekind proved every non-zero ideal factors uniquely into prime ideals.
The class group $\text{Cl}(K)$ is finite; its order $h_K$ measures failure of unique factorisation.
Minkowski's bound makes the class number effectively computable.
Dirichlet's Unit Theorem describes the group of units: rank $r_1 + r_2 - 1$ free part plus roots of unity.

References

BookNeukirch, J. Algebraic Number Theory. Springer, 1999.
BookMarcus, D. Number Fields. Springer, 1977.