21.02.07 · number-theory / quadratic-forms-local-fields

Number fields: ring of integers, ideal class group, and the Dirichlet unit theorem

shipped3 tiersLean: none

Anchor (Master): Neukirch 1999 *Algebraic Number Theory* (Springer Grundlehren 322) Ch. I (ring of integers, Dedekind domains, Minkowski theory, the class number and unit theorem); Lang 1994 *Algebraic Number Theory* (Springer GTM 110) Ch. V-VI; Dirichlet 1846 (unit theorem); Minkowski 1896 *Geometrie der Zahlen* (geometry of numbers)

Intuition Beginner

The whole numbers have one feature so familiar that it is easy to forget it is special: every whole number factors into primes in exactly one way. The number $12$ is $2 \times 2 \times 3$ , and there is no other prime recipe for it. This is the backbone of elementary number theory.

When you build a larger number system by adjoining a new number — say $- 5$ , so that you now have all the combinations $a + b - 5$ with $a, b$ whole — this clean property can break. In that system the number $6$ has two genuinely different prime recipes. Factorisation, the bedrock of the integers, stops being unique.

The repair is one of the great ideas of the nineteenth century. Instead of factoring numbers, you factor something slightly larger called an ideal. At the level of ideals, unique factorisation comes back perfectly. The price is that some ideals are not just "a single number in disguise". The collection of these stubborn ideals forms a group, and the size of that group — the class number — counts precisely how badly ordinary factorisation failed.

So the story has a slogan: in number systems beyond the integers, factorisation can fail, the class group counts by how much, and a piece of geometry shows the failure is always finite.

Visual Beginner

Picture the new number system as a grid of dots in the plane — a lattice. Each dot is one of the elements $a + b - 5$ , placed at the point with horizontal coordinate $a$ and vertical coordinate $b 5$ . The grid is perfectly regular, like graph paper stretched in one direction.

Two questions about the number system become questions about this grid. First, how spread out are the dots near the origin? A theorem of Minkowski says that any symmetric blob centred at the origin, once its area passes a fixed threshold, must swallow a grid dot other than the centre. That single geometric fact is what forces the class number to be a finite count rather than an endless one.

Second, which dots are units — the elements you can divide by, the analogues of $+ 1$ and $- 1$ ? For the grid drawn from $- 5$ there are only four. For a grid drawn from $2$ there are endlessly many, marching off to infinity along a curve. Counting which is which is the unit theorem.

Worked example Beginner

Take the number system $Z [- 5]$ , the dots $a + b - 5$ with $a, b$ whole. We show that $6$ has two different factorisations.

Recipe one is plain: $6 = 2 \times 3$ . Recipe two uses the new number: $(1 + - 5) (1 - - 5) = 1 - 5 \times (- 1)$ , and since $- 5$ squared is $- 5$ , this works out to $1 + 5 = 6$ . So $6 = (1 + - 5) (1 - - 5)$ as well.

Are these recipes really different, or is $2$ secretly a product involving $1 + - 5$ ? To check, attach to each element $a + b - 5$ the size $a^{2} + 5 b^{2}$ . This size multiplies: the size of a product is the product of the sizes. The size of $2$ is $4$ , the size of $3$ is $9$ , and the size of $1 \pm - 5$ is $1 + 5 = 6$ .

If $2$ split into a product of two non-unit pieces, their sizes would multiply to $4$ , forcing one piece to have size $2$ . But $a^{2} + 5 b^{2} = 2$ has no whole solution: $b$ must be $0$ , and then $a^{2} = 2$ is impossible. So $2$ cannot be broken further, and likewise $3$ and $1 \pm - 5$ each resist splitting.

Each of the four numbers $2, 3, 1 + - 5, 1 - - 5$ is unbreakable, yet $6$ is a product of the first two and also of the last two. Unique factorisation has failed. The class group of this system has size two, the smallest possible measure of a genuine failure.

Check your understanding Beginner

Exercise (easy, multiple choice).

What does the class number $h_{K}$ of a number field measure?

A. How many primes the field contains B. How far the ring of integers is from having unique factorisation of elements C. The number of units in the ring of integers D. The degree of the field over the rational numbers

Hint

The class number is the size of the ideal class group. When it equals one, the ring of integers is a unique factorisation domain.

Answer

B. How far the ring of integers is from having unique factorisation of elements. The class number is the size of the ideal class group $Cl (K)$ . It equals one exactly when the ring of integers is a principal ideal domain, which for these rings is the same as being a unique factorisation domain. A larger class number signals a larger failure of element-level unique factorisation.

Formal definition Intermediate+

A number field is a field $K$ that is a finite extension of $Q$ . Its degree is $n = [K : Q]$ , and there are exactly $n$ field embeddings $K ↪ C$ . These split into $r_{1}$ embeddings with image in $R$ and $2 r_{2}$ embeddings coming in complex-conjugate pairs, so that $n = r_{1} + 2 r_{2}$ . The pair $(r_{1}, r_{2})$ is the signature of $K$ .

An element $α \in K$ is an algebraic integer if it is a root of a monic polynomial with coefficients in $Z$ . The set of all algebraic integers in $K$ forms a subring $O_{K}$ , the ring of integers of $K$ ; it is the integral closure of $Z$ in $K$ . As an abelian group $O_{K}$ is free of rank $n$ , so it has an integral basis $ω_{1}, \dots, ω_{n}$ , and the determinant $d_{K} = det (σ_{i} (ω_{j}))^{2}$ over the embeddings $σ_{i}$ is the discriminant of $K$ , a non-zero integer.

The ring $O_{K}$ is a Dedekind domain: it is Noetherian, it is integrally closed in $K$ , and every non-zero prime ideal is maximal (Krull dimension one). The defining structural consequence is unique factorisation of ideals: every non-zero ideal $a \subseteq O_{K}$ factors as a product of prime ideals, $$ \mathfrak{a} = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_g^{e_g}, $$ and the primes $p_{i}$ with their exponents $e_{i} \geq 1$ are determined by $a$ .

A fractional ideal is a non-zero finitely generated $O_{K}$ -submodule of $K$ ; the fractional ideals form an abelian group $I_{K}$ under multiplication, with the principal fractional ideals $P_{K} = {x O_{K} : x \in K^{\times}}$ as a subgroup. The ideal class group is the quotient $$ \mathrm{Cl}(K) = \mathcal{I}_K / \mathcal{P}_K, $$ and the class number is $h_{K} = ∣ Cl (K) ∣$ . The ring $O_{K}$ is a principal ideal domain, equivalently a unique factorisation domain, exactly when $h_{K} = 1$ .

The norm of a non-zero ideal is $N (a) = ∣ O_{K} / a ∣$ , a positive integer, and it is multiplicative: $N (ab) = N (a) N (b)$ . The unit group $O_{K}^{\times}$ consists of the elements of $O_{K}$ whose inverse also lies in $O_{K}$ ; equivalently, those of ideal norm one. The torsion subgroup of $O_{K}^{\times}$ is the finite cyclic group $μ_{K}$ of roots of unity in $K$ .

Counterexamples to common slips

The ring of integers is the integral closure of $Z$ , not simply $Z [α]$ for a generator $α$ . For $K = Q (5)$ the ring of integers is $Z [\frac{1 + 5}{2}]$ , strictly larger than $Z [5]$ , because the golden-ratio element is a root of $x^{2} - x - 1$ and so is an algebraic integer.
A non-principal ideal is not "a number that fails to exist". The ideal $(2, 1 + - 5)$ in $Z [- 5]$ is a perfectly definite set; it simply cannot be written as $x O_{K}$ for a single $x$ . Its square is principal, which is why the class has order two.
The class number measures element factorisation, not ideal factorisation. Ideal factorisation is always unique in a Dedekind domain. The class group records how far the ideal-level uniqueness is from descending to elements.

Key theorem with proof Intermediate+

Theorem (finiteness of the class number; Minkowski, Dedekind). Let $K$ be a number field of degree $n = r_{1} + 2 r_{2}$ and discriminant $d_{K}$ . Then the class number $h_{K}$ is finite. Every ideal class contains an integral ideal $a$ whose norm satisfies the Minkowski bound $$ N(\mathfrak{a}) \leq M_K := \left( \frac{4}{\pi} \right)^{r_2} \frac{n!}{n^n} \sqrt{|d_K|}. $$

Proof. Embed $K$ into the Minkowski space $K_{R} = R^{r_{1}} \times C^{r_{2}} ≅ R^{n}$ by sending $α$ to the tuple of its real embeddings and one representative from each complex-conjugate pair. Under this embedding $O_{K}$ maps to a full lattice $Λ$ whose covolume is $covol (Λ) = 2^{- r_{2}} ∣ d_{K} ∣$ . More generally a non-zero ideal $a$ maps to a lattice of covolume $2^{- r_{2}} N (a) ∣ d_{K} ∣$ .

We use Minkowski's convex-body theorem: a centrally symmetric convex set $S \subseteq R^{n}$ with volume $vol (S) > 2^{n} covol (Λ)$ contains a lattice point other than the origin. Choose for $S$ a region adapted to the norm — concretely, the set where $\sum ∣ x_{i} ∣ + 2 \sum ∣ z_{j} ∣ \leq t$ — whose volume is computed to be $2^{r_{1}} (π /2)^{r_{2}} t^{n} / n!$ . Selecting $t$ so the volume just exceeds the convex-body threshold for the lattice attached to $a^{- 1}$ produces a non-zero element $α \in a^{- 1}$ with all embeddings small.

The arithmetic-geometric mean inequality bounds the absolute norm of that $α$ in terms of $t$ , and after optimising one obtains a non-zero $α \in a^{- 1}$ with $∣ N_{K / Q} (α) ∣ \leq M_{K} / N (a)$ . The integral ideal $b = α a$ then lies in the same ideal class as $a^{- 1}$ inverse-shifted and has norm $N (b) = ∣ N_{K / Q} (α) ∣ N (a) \leq M_{K}$ . So every ideal class has a representative of norm at most $M_{K}$ .

Finally there are only finitely many integral ideals of norm at most $M_{K}$ : an ideal of norm $m$ divides $m O_{K}$ , and a fixed integer $m$ has finitely many ideal divisors. Hence finitely many ideal classes, and $h_{K} < \infty$ . $□$

Bridge. This finiteness result builds toward the entire analytic theory of number fields, and the foundational reason it holds is exactly the conversion of an arithmetic question — how many ideal classes are there — into a lattice-point count, where Minkowski's convex-body theorem supplies a guaranteed small element. The central insight is that the ring of integers, sitting inside Minkowski space as a lattice of covolume $2^{- r_{2}} ∣ d_{K} ∣$ , makes the discriminant a geometric volume, so that the size of $∣ d_{K} ∣$ controls both how rare small elements are and how large the class number can be. This is exactly the mechanism that appears again in 21.03.03, where the class number $h_{K}$ , the regulator $R_{K}$ , the root-of-unity count $w_{K}$ , and the discriminant assemble into the residue of the Dedekind zeta function. The bridge is the recognition that geometry of numbers is dual to analytic counting: the convex-body theorem and the zeta residue are two readings of the same lattice. Putting these together, the finiteness of $h_{K}$ generalises into the broader principle that arithmetic invariants of $K$ are encoded in the geometry of its Minkowski lattice, a pattern that recurs from the unit theorem to Arakelov theory.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

In $Z [- 5]$ verify that $p^{2} = (2)$ where $p = (2, 1 + - 5)$ , and explain why this shows $p$ has order two in the class group.

Hint

Expand $p^{2}$ as the ideal generated by all products of the two generators, then reduce using $(1 + - 5)^{2} = - 4 + 2 - 5$ .

Answer

$p^{2} = (2)$ , so $[p]$ has order two. The ideal $p^{2}$ is generated by $4$ , $2 (1 + - 5)$ , and $(1 + - 5)^{2} = - 4 + 2 - 5$ . Every generator is a multiple of $2$ : $4 = 2 \cdot 2$ , $2 (1 + - 5) = 2 (1 + - 5)$ , and $- 4 + 2 - 5 = 2 (- 2 + - 5)$ . Conversely $2 = 4 - 2 (1 + - 5) + (1 + - 5)$ adjusted lies in $p^{2}$ since $4$ and $- 4 + 2 - 5$ combine to give $2 \cdot$ (a unit-free combination), and a direct check gives $2 \in p^{2}$ . Hence $p^{2} = (2)$ , a principal ideal, so $[p]^{2} = [(2)] = 1$ in $Cl (K)$ . Because $p$ itself is not principal (no element has norm $2$ , as $a^{2} + 5 b^{2} = 2$ is unsolvable), $[p] \neq = 1$ , and the order is exactly two. Rubric: full credit for $p^{2} = (2)$ and the order-two conclusion.

Exercise 6 (hard, short-answer).

State the Dirichlet unit theorem and use it to predict the structure of $O_{K}^{\times}$ for a real quadratic field, an imaginary quadratic field, and a totally real cubic field.

Hint

The free rank is $r_{1} + r_{2} - 1$ . Determine the signature in each case, then read off the rank and the torsion (the roots of unity).

Answer

Theorem. For a number field $K$ of signature $(r_{1}, r_{2})$ , $O_{K}^{\times} ≅ μ_{K} \times Z^{r_{1} + r_{2} - 1}$ , where $μ_{K}$ is the finite cyclic group of roots of unity in $K$ . Real quadratic ( $r_{1} = 2$ , $r_{2} = 0$ ): rank $2 + 0 - 1 = 1$ , and the only roots of unity are $\pm 1$ , so $O_{K}^{\times} ≅ {\pm 1} \times Z$ , generated by $\pm 1$ and a fundamental unit. Imaginary quadratic ( $r_{1} = 0$ , $r_{2} = 1$ ): rank $0 + 1 - 1 = 0$ , so the unit group is just $μ_{K}$ , finite; it is ${\pm 1}$ except for $Q (i)$ (four units) and $Q (- 3)$ (six units). Totally real cubic ( $r_{1} = 3$ , $r_{2} = 0$ ): rank $3 + 0 - 1 = 2$ , with roots of unity $\pm 1$ , so $O_{K}^{\times} ≅ {\pm 1} \times Z^{2}$ , two independent fundamental units. Rubric: full credit for the theorem statement and all three correct rank-and-torsion structures.

Exercise 7 (hard, short-answer).

Use the Minkowski bound to prove that $Q (- 2)$ has class number one.

Hint

Here $d_{K} = - 8$ , $n = 2$ , $r_{2} = 1$ . Compute $M_{K}$ , then check which prime ideals of norm below the bound exist and whether they are principal.

Answer

$h_{K} = 1$ . The discriminant is $d_{K} = - 8$ , so $M_{K} = (2/ π) 8 = (42) / π \approx 1.80$ . Every ideal class has a representative of norm at most $1$ , since the only integer norm below $1.80$ that an integral ideal can carry — apart from $1$ — would be $1$ itself once we note no ideal has norm strictly between. The only ideal of norm $1$ is $O_{K}$ , the identity class. (To be precise the bound permits norm $1$ only, because a norm- $1$ ideal is the whole ring; were the bound to reach $2$ , we would check the prime above $2$ , but here it does not.) Hence the class group is the identity group and $h_{K} = 1$ ; $Z [- 2]$ is a principal ideal domain. Rubric: full credit for $M_{K} \approx 1.80$ and the conclusion that only the identity class survives.

Advanced results Master

Theorem (Dirichlet unit theorem; Dirichlet 1846). Let $K$ be a number field of signature $(r_{1}, r_{2})$ and set $r = r_{1} + r_{2} - 1$ . Then the unit group is $$ \mathcal{O}_K^\times \cong \mu_K \times \mathbb{Z}^{r}, $$ where $μ_{K}$ is the finite cyclic group of roots of unity in $K$ . A choice of $r$ units generating the free part is a system of fundamental units.

The proof embeds the units logarithmically. Define $L : O_{K}^{\times} \to R^{r_{1} + r_{2}}$ by $L (ε) = (lo g ∣ σ_{1} (ε) ∣, \dots, lo g ∣ σ_{r_{1}} (ε) ∣, 2 lo g ∣ τ_{1} (ε) ∣, \dots, 2 lo g ∣ τ_{r_{2}} (ε) ∣)$ over the real embeddings $σ_{i}$ and complex pairs $τ_{j}$ . Because $∣ N_{K / Q} (ε) ∣ = 1$ for a unit, the image lies in the trace-zero hyperplane $H = {\sum x_{i} = 0}$ , of dimension $r$ . The kernel of $L$ is the finite group $μ_{K}$ . The substance is that $L (O_{K}^{\times})$ is a full lattice in $H$ : it is discrete (a bounded region of $H$ pulls back to finitely many units, by the finiteness of integers with bounded embeddings) and it spans $H$ (proved by a Minkowski-style convex-body argument producing units with prescribed sign pattern of logarithms). A full lattice in $R^{r}$ is isomorphic to $Z^{r}$ , giving the free rank $r$ .

Definition (regulator). Discard one coordinate from $L$ (the hyperplane $H$ has dimension $r$ ) and let $ε_{1}, \dots, ε_{r}$ be fundamental units. The regulator $R_{K} = ∣ det (λ_{i} (ε_{j}))_{1 \leq i, j \leq r} ∣$ is the covolume of the unit lattice $L (O_{K}^{\times})$ in $H$ , where $λ_{i}$ are the first $r$ logarithmic coordinates. It is a positive real number independent of the choice of fundamental units and of the discarded coordinate.

Theorem (analytic class-number formula; statement). The Dedekind zeta function $ζ_{K} (s)$ has a simple pole at $s = 1$ with residue $$ \operatorname{res}_{s=1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} , h_K , R_K}{w_K \sqrt{|d_K|}}, $$ where $w_{K} = ∣ μ_{K} ∣$ . This single identity bundles every invariant produced in this unit — the class number $h_{K}$ , the regulator $R_{K}$ , the root-of-unity count $w_{K}$ , the discriminant $d_{K}$ , and the signature $(r_{1}, r_{2})$ — and the consumer unit 21.03.03 develops it in full.

Theorem (ramification and the discriminant). A prime $p$ ramifies in $K$ — meaning $p O_{K} = \prod p_{i}^{e_{i}}$ has some $e_{i} \geq 2$ — if and only if $p ∣ d_{K}$ . Only finitely many primes ramify, exactly the divisors of the discriminant. For $Q (- 5)$ with $d_{K} = - 20$ the ramified primes are $2$ and $5$ , matching $p = (2, 1 + - 5)$ with $p^{2} = (2)$ and the prime above $5$ with $(- 5)^{2} = (5)$ .

Theorem (Hermite-Minkowski finiteness). For each bound $B$ and degree $n$ , there are only finitely many number fields $K$ of degree $n$ with $∣ d_{K} ∣ \leq B$ . Moreover $∣ d_{K} ∣ > 1$ for every $K \neq = Q$ (Minkowski), so $Q$ is the unique number field of discriminant $\pm 1$ . The Minkowski bound forces $∣ d_{K} ∣ \geq (π /4)^{r_{2}} n^{n} / n!$ , which exceeds $1$ for $n \geq 2$ ; this is the same convex-body inequality, now read as a lower bound on the discriminant rather than an upper bound on a class representative.

Synthesis. The arithmetic of a number field is governed by three finiteness statements that are one statement seen from different sides, and the foundational reason they cohere is exactly Minkowski's convex-body theorem applied to the ring of integers as a lattice. The central insight is that $O_{K}$ sits in Minkowski space with covolume $2^{- r_{2}} ∣ d_{K} ∣$ , so the discriminant is a geometric volume; the convex-body theorem then forces small elements to exist, and this single mechanism is dual to all three results — the finiteness of the class number, the full-lattice property of the logarithmic unit embedding, and the lower bound $∣ d_{K} ∣ > 1$ . Putting these together, the class number $h_{K}$ and the regulator $R_{K}$ are the order and the covolume of two lattices attached to $K$ , and this is exactly the pairing that the analytic class-number formula reads off the residue of $ζ_{K}$ at $s = 1$ . The bridge is the recognition that geometry of numbers, the ideal class group, and the unit group are three faces of the Minkowski lattice of $O_{K}$ ; this pattern recurs in Arakelov geometry, where the same lattice acquires a metric and the class-number formula becomes an arithmetic Riemann-Roch statement, and it generalises into the adelic picture where the class group and unit group reappear as the components of the idele class group.

Full proof set Master

Proposition 1 (the ring of integers is a free $Z$ -module of rank $n$ ). Let $K$ be a number field of degree $n$ . Then $O_{K}$ is a free abelian group of rank $n$ .

Proof. Pick a $Q$ -basis $α_{1}, \dots, α_{n}$ of $K$ consisting of algebraic integers (clear denominators in any basis). The trace form $T (x, y) = Tr_{K / Q} (x y)$ is a non-degenerate symmetric bilinear form, because $K / Q$ is separable. Let $β_{1}, \dots, β_{n}$ be the dual basis under $T$ . Any $γ \in O_{K}$ expands as $γ = \sum c_{i} β_{i}$ with $c_{i} = T (γ, α_{i}) = Tr_{K / Q} (γ α_{i}) \in Z$ , since traces of algebraic integers are rational integers. Hence $$ \mathbb{Z}\alpha_1 + \cdots + \mathbb{Z}\alpha_n \subseteq \mathcal{O}_K \subseteq \mathbb{Z}\beta_1 + \cdots + \mathbb{Z}\beta_n. $$ The outer groups are free of rank $n$ , and a subgroup of a free abelian group of rank $n$ containing a finite-index free subgroup of rank $n$ is itself free of rank $n$ (structure theorem for finitely generated abelian groups). So $O_{K}$ is free of rank $n$ . $□$

Proposition 2 (unique factorisation of ideals). In a Dedekind domain $O_{K}$ , every non-zero ideal factors uniquely into prime ideals.

Proof. First, every non-zero ideal contains a product of non-zero prime ideals: if not, the set of ideals failing this is non-empty, and by the Noetherian property has a maximal element $a$ , which cannot be prime, so there are $x, y \in / a$ with $x y \in a$ ; then $a + (x)$ and $a + (y)$ strictly contain $a$ , each contains a product of primes, and so does their product, which lies in $a$ — a contradiction. Next, for a non-zero prime $p$ define $p^{- 1} = {x \in K : x p \subseteq O_{K}}$ ; using that $O_{K}$ is integrally closed one shows $p^{- 1}$ strictly contains $O_{K}$ and $p p^{- 1} = O_{K}$ , so every non-zero prime is invertible. Existence of factorisation now follows by Noetherian induction: a maximal ideal $p \supseteq a$ gives $a \subseteq a p^{- 1} \subseteq O_{K}$ with $a p^{- 1}$ strictly larger, and iterating terminates. Uniqueness follows because if $p_{1} \dots p_{g} = q_{1} \dots q_{h}$ then $p_{1}$ contains the right side, hence contains some $q_{j}$ , hence equals it by maximality; cancel using invertibility and induct. $□$

Proposition 3 (the norm is multiplicative and finite). For non-zero ideals $a, b$ of $O_{K}$ , the index $N (a) = ∣ O_{K} / a ∣$ is finite and $N (ab) = N (a) N (b)$ .

Proof. Finiteness: a non-zero ideal contains a non-zero rational integer $m$ (the absolute norm of any non-zero element), so $O_{K} / a$ is a quotient of $O_{K} / m O_{K} ≅ (Z / m)^{n}$ , a finite group. Multiplicativity reduces by unique factorisation to the case $b = p$ prime, where the third isomorphism theorem gives $∣ O_{K} / ap ∣ = ∣ O_{K} / a ∣ \cdot ∣ a / ap ∣$ , and $a / ap$ is a one-dimensional vector space over the residue field $O_{K} / p$ (using invertibility of $p$ to see the quotient is $O_{K} / p$ as a module), so $∣ a / ap ∣ = N (p)$ . Hence $N (ap) = N (a) N (p)$ and the general case follows by induction. $□$

Proposition 4 (Minkowski's convex-body theorem). Let $Λ \subseteq R^{n}$ be a full lattice and $S \subseteq R^{n}$ a centrally symmetric convex set with $vol (S) > 2^{n} covol (Λ)$ . Then $S$ contains a lattice point other than the origin.

Proof. Consider the half-scaled set $\frac{1}{2} S$ , of volume $2^{- n} vol (S) > covol (Λ)$ . The quotient map $R^{n} \to R^{n} /Λ$ restricted to a set of volume exceeding the covolume of $Λ$ cannot be injective (the target torus has volume $covol (Λ)$ ), so there are distinct $x, y \in \frac{1}{2} S$ with $x - y \in Λ ∖ {0}$ . Write $x - y = \frac{1}{2} (2 x) + \frac{1}{2} (- 2 y)$ ; both $2 x$ and $- 2 y$ lie in $S$ (using central symmetry for $- 2 y$ ), and convexity puts their midpoint $x - y$ in $S$ . So $x - y$ is a non-zero lattice point in $S$ . $□$

Proposition 5 (no non-zero element of a non-principal class has too small a norm — finiteness recap). Every ideal class of $K$ contains an integral ideal of norm at most $M_{K} = (4/ π)^{r_{2}} (n! / n^{n}) ∣ d_{K} ∣$ , and consequently $h_{K}$ is finite.

Proof. Given a class $c$ , pick an integral ideal $a$ in the inverse class $c^{- 1}$ . Apply Proposition 4 in Minkowski space to the lattice attached to $a$ and the norm-adapted convex body of volume just above the threshold; this yields a non-zero $α \in a$ with $∣ N_{K / Q} (α) ∣ \leq M_{K} \cdot N (a)$ , after the arithmetic-geometric mean estimate on the embeddings. The ideal $b = (α) a^{- 1}$ is integral, lies in $c$ , and has $N (b) = ∣ N_{K / Q} (α) ∣/ N (a) \leq M_{K}$ . Since only finitely many ideals have norm at most $M_{K}$ — each divides the principal ideal generated by its norm, and a fixed integer has finitely many ideal divisors — there are finitely many classes. $□$

Proposition 6 ( $Q (- 5)$ has class number $2$ ). The field $K = Q (- 5)$ satisfies $h_{K} = 2$ , with the class group generated by $[p]$ , $p = (2, 1 + - 5)$ .

Proof. Here $n = 2$ , $r_{2} = 1$ , $d_{K} = - 20$ , so $M_{K} = (4/ π) (2! /4) 20 = (2/ π) 20 \approx 2.85$ . Every class has a representative of norm at most $2$ . Norm- $1$ ideals are the whole ring (the identity class). For norm $2$ : the prime $2$ factors as $p^{2}$ with $p = (2, 1 + - 5)$ of norm $2$ (Exercise 3), the unique ideal of norm $2$ . This $p$ is not principal, since a generator would have norm $2$ , but $a^{2} + 5 b^{2} = 2$ has no integer solution. So $[p] \neq = 1$ , while $[p]^{2} = [p^{2}] = [(2)] = 1$ . The class group is generated by classes of norm $\leq 2$ ideals, namely the identity and $[p]$ , so $Cl (K) = {1, [p]} ≅ Z /2$ and $h_{K} = 2$ . $□$

Connections Master

Unique factorisation in $Z$ 21.01.02. The fundamental theorem of arithmetic is the special case $K = Q$ , where $O_{K} = Z$ , every ideal is principal, the class number is one, and prime ideals are the prime numbers. This unit is the precise statement of what survives and what fails when $Z$ is enlarged to $O_{K}$ : ideal factorisation always survives, element factorisation survives only when $h_{K} = 1$ . The class group is the exact obstruction, so the entire subject is the controlled generalisation of the result that primes factor numbers uniquely.
Polynomial-ring PID/UFD/Euclidean hierarchy 01.02.07. The chain Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD organises the failure measured here. The ring $O_{K}$ is a Dedekind domain, and a Dedekind domain is a PID if and only if it is a UFD if and only if $h_{K} = 1$ — the equivalence of PID and UFD, which can fail for general rings, holds for rings of integers precisely because the class group controls both. Integral dependence over $Z$ , the algebraic backbone of $O_{K}$ as the integral closure, is the ring-theoretic content imported from that unit.
$p$ -adic numbers and completions 21.02.03. Localising and completing $O_{K}$ at a prime $p$ above $p$ produces a local field, and the factorisation $p O_{K} = \prod p_{i}^{e_{i}}$ records the ramification and residue degrees that the local theory reads off. The global class group and unit group are assembled from local data through the idele class group, the local-global bridge: the geometry-of-numbers finiteness here is the global shadow of the local compactness exploited in the completions, and the two viewpoints meet in the analytic class-number formula.
Dedekind, Hecke, and Artin $L$ -functions 21.03.03. This is the consumer. The Dedekind zeta function $ζ_{K} (s)$ has a simple pole at $s = 1$ whose residue is $2^{r_{1}} (2 π)^{r_{2}} h_{K} R_{K} / (w_{K} ∣ d_{K} ∣)$ , packaging exactly the five invariants constructed here — class number, regulator, root-of-unity count, discriminant, and signature. The class-number-formula proof route cites the finiteness of $h_{K}$ and the Dirichlet unit theorem as named black boxes; this unit supplies them, closing a forward reference that several later units in the chapter depend on.
Heights and the Northcott property 21.09.03. The Northcott finiteness property — only finitely many points of bounded height and bounded degree — is the Diophantine-geometry analogue of the Minkowski finiteness proved here: both convert a boundedness condition into a finite count via a lattice or a counting estimate. The regulator of this unit reappears as the height regulator in the Birch and Swinnerton-Dyer setting, and the convex-body mechanism that bounds class representatives is the model for the height machine that bounds rational points.

Historical & philosophical context Master

The subject was born from a failure. In the 1840s Kummer, studying Fermat's Last Theorem in cyclotomic fields $Q (ζ_{p})$ , discovered that unique factorisation of elements collapses, and he rescued the theory by inventing "ideal numbers" — fictitious common divisors that restore unique factorisation. Dedekind's Supplement X to Dirichlet's Vorlesungen über Zahlentheorie (1871) ^{[Dedekind 1871]} replaced Kummer's ideal numbers with honest sets — ideals — and proved that in the ring of integers of any number field, ideals factor uniquely into prime ideals. This is the conceptual leap that defines modern algebraic number theory: the right objects to factor are ideals, and the class group $Cl (K) = I_{K} / P_{K}$ is the exact measure of how far ideal factorisation is from element factorisation. Dedekind's notion of a Dedekind domain — Noetherian, integrally closed, dimension one — abstracts the three properties that make this work, and it became one of the foundational structures of commutative algebra.

Dirichlet's 1846 paper Zur Theorie der complexen Einheiten ^{[Dirichlet 1846]} settled the structure of the unit group, proving that the units of $O_{K}$ form a group of the shape $μ_{K} \times Z^{r_{1} + r_{2} - 1}$ . The rank formula encodes the signature of the field directly: real and complex places contribute differently, and the fundamental units that generate the free part are, for real quadratic fields, exactly the solutions of the Pell equation studied since antiquity. Minkowski's Geometrie der Zahlen (1896) ^{[Minkowski 1896]} supplied the geometric engine. His convex-body theorem — a symmetric convex region of large enough volume must contain a lattice point — turned arithmetic finiteness into lattice-point counting, proving the finiteness of the class number and giving the explicit Minkowski bound on class representatives. The same theorem yields $∣ d_{K} ∣ > 1$ for every field beyond $Q$ , so $Q$ is recognised as the unique unramified-everywhere number field.

Philosophically, the arc from Kummer to Minkowski is the discovery that the obstructions to naive arithmetic are themselves structured objects — a finite group and a lattice — carrying their own invariants. The class number and the regulator are not defects to be apologised for but invariants to be computed, and Dirichlet's analytic class-number formula reveals that they are encoded in the analytic behaviour of the Dedekind zeta function. This fusion of algebra, geometry, and analysis in a single field of numbers set the template for the twentieth century: class field theory, the geometry of numbers, and ultimately the Langlands programme all descend from the recognition that the arithmetic of a number field is governed by the geometry of its lattice of integers.

Bibliography Master

@incollection{Dedekind1871,
  author    = {Dedekind, Richard},
  title     = {{\"U}ber die Theorie der ganzen algebraischen Zahlen},
  booktitle = {Vorlesungen {\"u}ber Zahlentheorie von P. G. L. Dirichlet (Supplement X)},
  edition   = {2},
  publisher = {Vieweg},
  address   = {Braunschweig},
  year      = {1871}
}

@article{Dirichlet1846,
  author  = {Dirichlet, Peter Gustav Lejeune},
  title   = {Zur Theorie der complexen Einheiten},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {30},
  year    = {1846},
  pages   = {103--107}
}

@book{Minkowski1896,
  author    = {Minkowski, Hermann},
  title     = {Geometrie der Zahlen},
  publisher = {Teubner},
  address   = {Leipzig},
  year      = {1896}
}

@book{Neukirch1999,
  author    = {Neukirch, J{\"u}rgen},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {322},
  year      = {1999},
  note      = {Translated by Norbert Schappacher}
}

@book{Marcus1977,
  author    = {Marcus, Daniel A.},
  title     = {Number Fields},
  publisher = {Springer-Verlag},
  series    = {Universitext},
  year      = {1977}
}

@book{Lang1994,
  author    = {Lang, Serge},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {110},
  edition   = {2},
  year      = {1994}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2},
  year      = {2005}
}

@book{Serre1973,
  author    = {Serre, Jean-Pierre},
  title     = {A Course in Arithmetic},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {7},
  year      = {1973},
  note      = {English translation of \emph{Cours d'arithm{\'e}tique}, Presses Universitaires de France, 1970}
}

@book{StewartTall2015,
  author    = {Stewart, Ian and Tall, David},
  title     = {Algebraic Number Theory and Fermat's Last Theorem},
  publisher = {CRC Press},
  edition   = {4},
  year      = {2015}
}

@book{Samuel1970,
  author    = {Samuel, Pierre},
  title     = {Algebraic Theory of Numbers},
  publisher = {Hermann},
  address   = {Paris},
  year      = {1970}
}

Prerequisites

21.01.02
01.02.07
21.02.03

Tier anchors

beginner: Ireland-Rosen 1990 *A Classical Introduction to Modern Number Theory* (Springer GTM 84) Ch. 12 informal opening; Stewart-Tall 2015 *Algebraic Number Theory and Fermat's Last Theorem* Ch. 1 informal opening
intermediate: Marcus 1977 *Number Fields* (Springer) Ch. 1-3, 5; Neukirch 1999 *Algebraic Number Theory* (Springer Grundlehren 322) Ch. I §§1-7
master: Neukirch 1999 *Algebraic Number Theory* (Springer Grundlehren 322) Ch. I (ring of integers, Dedekind domains, Minkowski theory, the class number and unit theorem); Lang 1994 *Algebraic Number Theory* (Springer GTM 110) Ch. V-VI; Dirichlet 1846 (unit theorem); Minkowski 1896 *Geometrie der Zahlen* (geometry of numbers)

References

Dedekind, R. — Über die Theorie der ganzen algebraischen Zahlen (Supplement X to Dirichlet's Vorlesungen über Zahlentheorie) · Supplement X to P. G. L. Dirichlet, *Vorlesungen über Zahlentheorie*, 2nd ed., Vieweg (Braunschweig, 1871), and expanded in the 4th ed. (1894). The originator text for ideals, prime-ideal factorisation in the ring of integers, the Dedekind domain, and the ideal class group as the modern replacement for Kummer's ideal numbers.
Dirichlet, P. G. L. — Zur Theorie der complexen Einheiten · *Journal für die reine und angewandte Mathematik* (Crelle) 30 (1846), based on the 1846 Berlin Academy report. The originator paper for the unit theorem: the group of units of the ring of integers of a number field is the product of the finite group of roots of unity and a free abelian group of rank $r_1 + r_2 - 1$.
Minkowski, H. — Geometrie der Zahlen · Teubner (Leipzig, 1896; reprinted Chelsea 1953). The originator monograph for the geometry of numbers: the convex-body theorem that a centrally symmetric convex set of sufficiently large volume contains a non-zero lattice point, and its application to the finiteness of the class number and to the unit theorem via the logarithmic embedding.
Neukirch, J. — Algebraic Number Theory · Springer Grundlehren der mathematischen Wissenschaften 322 (1999), translated by N. Schappacher. Ch. I §§1-7 develops integrality, the ring of integers, Dedekind domains and ideal factorisation, the Minkowski theory (lattices, the convex-body theorem, the class-number finiteness), and the Dirichlet unit theorem with the regulator, in the canonical modern presentation.
Marcus, D. A. — Number Fields · Springer Universitext (1977; 2nd ed. 2018). Ch. 1-5 give a concrete, example-driven development of the ring of integers, prime-ideal factorisation, the class group, the Minkowski bound, and the unit theorem, with extensive quadratic- and cubic-field computations and the canonical exercise sets.
Lang, S. — Algebraic Number Theory · Springer Graduate Texts in Mathematics 110 (2nd ed., 1994). Ch. I-V cover Dedekind rings, the geometry of numbers, the finiteness of the class number, and the unit theorem at graduate-text depth, including the lattice-point proof of Minkowski's theorem and the regulator.
Manin, Yu. I. and Panchishkin, A. A. — Introduction to Modern Number Theory · Springer, Encyclopaedia of Mathematical Sciences 49 (2nd ed., 2005). Ch. 1 §§5-7 present the arithmetic of algebraic numbers — rings of integers, ideal factorisation, the class group, Minkowski's bound and finiteness, and the Dirichlet unit theorem — as the foundational layer beneath the book's later $L$-function and Galois-representation material.
Serre, J.-P. — A Course in Arithmetic · Springer Graduate Texts in Mathematics 7 (1973), translation of *Cours d'arithmétique* (PUF 1970). Ch. IV-V treat quadratic fields, the structure of their rings of integers, and the worked computations of class numbers and units for imaginary and real quadratic fields used in the present unit's examples.
Stewart, I. and Tall, D. — Algebraic Number Theory and Fermat's Last Theorem · CRC Press (4th ed., 2015). Ch. 1-5 give the most elementary modern route into rings of integers, factorisation of ideals, the class group, and the Minkowski bound, motivated throughout by the failure of unique factorisation that obstructs naive proofs of Fermat's Last Theorem.

Estimated time

beginner: 18m
intermediate: 45m
master: 95m