Algebraic Number Theory: Foundations¶

Job: fix the basic geometry of number fields — integers, embeddings, discriminants, ideal theory, class group, and units — as used throughout arithmetic geometry and ANT.

Core¶

Number fields and embeddings¶

A number field $K$ is a finite extension of $\mathbb{Q}$, $n=[K:\mathbb{Q}]$. There are $n$ embeddings $K\hookrightarrow\mathbb{C}$: $r_1$ real and $2r_2$ complex (conjugate pairs), $r_1+2r_2=n$. Minkowski space: $$ K\otimes_\mathbb{Q}\mathbb{R}\cong\mathbb{R}^{r_1}\times\mathbb{C}^{r_2}\cong\mathbb{R}^n. $$

Ring of integers¶

$\alpha\in K$ is an algebraic integer if it satisfies a monic polynomial over $\mathbb{Z}$. The set $\mathcal{O}_K$ of algebraic integers in $K$ is a ring — the ring of integers of $K$. As a $\mathbb{Z}$-module, $\mathcal{O}_K$ is free of rank $n$; a $\mathbb{Z}$-basis is an integral basis.

Examples. $\mathcal{O}_{\mathbb{Q}}=\mathbb{Z}$; $\mathcal{O}_{\mathbb{Q}(i)}=\mathbb{Z}[i]$; for quadratic $K=\mathbb{Q}(\sqrt{d})$ ($d$ squarefree), $$ \mathcal{O}_K=\begin{cases}\mathbb{Z}\bigl[\frac{1+\sqrt{d}}{2}\bigr]&d\equiv 1\pmod 4,\ \mathbb{Z}[\sqrt{d}]&d\equiv 2,3\pmod 4.\end{cases} $$

Discriminant¶

For $\alpha_1,\ldots,\alpha_n\in K$, $$ \Delta(\alpha_1,\ldots,\alpha_n)=\det\bigl(\operatorname{Tr}_{K/\mathbb{Q}}(\alpha_i\alpha_j)\bigr)=\bigl(\det \sigma_k(\alpha_j)\bigr)^2. $$ Discriminant $d_K=\Delta(\omega_1,\ldots,\omega_n)$ for any integral basis $(\omega_j)$; independent of basis up to $(\mathbb{Z}^\times)^2$. $p\mid d_K$ $\Leftrightarrow$ $p$ ramifies in $K$ (Dedekind–Kummer / Dedekind theorem on different).

Unique factorization failure¶

$\mathcal{O}_K$ is always integrally closed, Noetherian, dimension $1$ — but not always a UFD or PID. Classic: $\mathbb{Z}[\sqrt{-5}]$, $6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$.

Dedekind domains and ideal factorization¶

A Dedekind domain is an integrally closed Noetherian domain of Krull dimension $\le 1$. Equivalently: every nonzero ideal factors uniquely as a product of prime ideals.

Theorem. $\mathcal{O}_K$ is a Dedekind domain. Every nonzero ideal $\mathfrak{a}\subset\mathcal{O}_K$ factors uniquely $$ \mathfrak{a}=\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_g^{e_g}. $$

Prime splitting. For $0\neq p\in\mathbb{Z}$, $p\mathcal{O}_K=\mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_g^{e_g}$ with $\sum_i e_i f_i=n$, $f_i=[\mathcal{O}_K/\mathfrak{p}_i:\mathbb{Z}/p\mathbb{Z}]$. Terminology: $e_i>1$ ramified; $g=n$ split completely; $g=1,e=1,f=n$ inert (in Galois case, $e,f$ constant and $efg=n$).

Fractional ideals. Finitely generated $K$-submodules $\mathfrak{a}\subset K$ with $d\mathfrak{a}\subset\mathcal{O}_K$ for some $d\neq 0$; form a group under multiplication. Principal fractional ideals $K^\times/\mathcal{O}_K^\times\hookrightarrow$ fractional ideals.

Ideal class group¶

Class group $\mathrm{Cl}_K=$ (fractional ideals) $/$ (principal fractional ideals). Equivalently $\mathrm{Pic}(\mathcal{O}_K)$.

Theorem (finiteness). $\mathrm{Cl}_K$ is a finite abelian group. Class number $h_K=|\mathrm{Cl}_K|$. $h_K=1\Leftrightarrow\mathcal{O}_K$ is a PID $\Leftrightarrow\mathcal{O}_K$ is a UFD.

Minkowski bound: every class contains an integral ideal of norm $\le M_K:=(n!/n^n)(4/\pi)^{r_2}\sqrt{|d_K|}$; finiteness follows by finiteness of ideals of bounded norm.

Dirichlet unit theorem¶

Theorem (Dirichlet). $$ \mathcal{O}_K^\times\cong \mu_K\times\mathbb{Z}^{r_1+r_2-1}, $$ where $\mu_K$ is the finite cyclic group of roots of unity in $K$. Unit rank $r=r_1+r_2-1$.

Log embedding. $\lambda:\mathcal{O}_K^\times\to\mathbb{R}^{r_1+r_2}$, $\lambda(u)=(\log|\sigma_i(u)|)_i$ lands in the hyperplane $\sum a_i x_i=0$ ($a_i=1$ or $2$); image is a lattice of full rank in that hyperplane. Regulator $R_K=$ covolume of the unit lattice.

Class number formula (pointer). Residue of $\zeta_K$ at $s=1$: $$ \operatorname{Res}_{s=1}\zeta_K(s)=\frac{2^{r_1}(2\pi)^{r_2}h_K R_K}{w_K\sqrt{|d_K|}}, $$ $w_K=|\mu_K|$. Links analytic NT to $h_K R_K$.

See it¶

Gaussian integers. $K=\mathbb{Q}(i)$, $\mathcal{O}_K=\mathbb{Z}[i]$, Euclidean $\Rightarrow$ PID, $h_K=1$, units $\{\pm 1,\pm i\}=\mu_4$, rank $r_1+r_2-1=0$. Odd prime $p$ splits iff $p\equiv 1\pmod 4$ (Fermat two-squares via $p=\pi\bar\pi$).

Real quadratic units. $K=\mathbb{Q}(\sqrt{2})$, units $\pm(1+\sqrt{2})^\mathbb{Z}$, fundamental unit $1+\sqrt{2}$, rank $1$. Norm $\pm 1$ Pell equation $x^2-2y^2=\pm 1$.

Check¶

Prove that if $\mathcal{O}_K$ is a PID then it is a UFD. Where does the converse use the Dedekind property?

PID $\Rightarrow$ UFD is general domain theory (irreducibles are prime). Conversely, in a Dedekind domain UFD $\Rightarrow$ every prime ideal is principal (take a prime element in $\mathfrak{p}$, use unique factorization of $(\pi)$) $\Rightarrow$ all ideals principal. Without Dedekind, UFD need not be PID (e.g. $k[x,y]$).

Compute $r_1,r_2$ and unit rank for $K=\mathbb{Q}(\sqrt[3]{2})$.

Minimal polynomial $x^3-2$: one real root, two complex $\Rightarrow$ $r_1=1$, $r_2=1$, $n=3$. Unit rank $1+1-1=1$.

Pitfalls¶

Not every order is Dedekind: non-maximal orders have non-invertible ideals; factorizations need the conductor.
Norm of ideals vs elements: $N(\mathfrak{a})=|\mathcal{O}_K/\mathfrak{a}|$; $N((\alpha))=|N_{K/\mathbb{Q}}(\alpha)|$. Signs of field norms matter for units.
Ramified primes divide $d_K$ only — converse true; unramified outside $d_K$.
Minkowski bound is rarely sharp for computation; use it to enumerate classes, then multiply ideals.

Refs¶

Neukirch, Algebraic Number Theory, Ch. I–III
Marcus, Number Fields
Lang, Algebraic Number Theory
Borevich–Shafarevich, Number Theory

Next¶

Algebraic NT overview · Class field theory · Number theory · Ring theory · Galois theory · Analytic NT