CM Fields, Class Towers & the Algebraic Toolkit

Overview

The unit distance proof bridges combinatorial geometry and algebraic number theory. This page documents the algebraic toolkit used in the construction: CM fields, class groups, genus theory, unramified extensions, and the critical Lemma 2.2 that generates unit-norm elements.

1. Number Fields and Extensions

1.1 Basic Definitions

A number field $K$ is a finite extension of $\mathbb{Q}$. Key invariants:

• Degree: $[K:\mathbb{Q}]$
• Discriminant: $\Delta_K$ (measures ramification)
• Ring of integers: $\mathcal{O}_K$
• Unit group: $\mathcal{O}_K^\times$ (Dirichlet's unit theorem)

1.2 Places and Embeddings

A number field $K$ of degree $n$ has $n$ embeddings into $\mathbb{C}$:

• $r_1$ real embeddings: $\sigma_i: K \hookrightarrow \mathbb{R}$
• $r_2$ pairs of complex conjugate embeddings: $\sigma_j, \bar{\sigma}_j: K \hookrightarrow \mathbb{C}$

with $r_1 + 2r_2 = n$.

1.3 Ramification

A prime $p$ ramifies in $K$ if $p$ divides the discriminant $\Delta_K$. In the ring of integers:

$p\mathcal{O}_K = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_g^{e_g}$

where $e_i > 1$ for at least one $i$. The prime is unramified if all $e_i = 1$.

2. CM Fields

2.1 Definition

A CM field (Complex Multiplication field) is a totally imaginary quadratic extension of a totally real number field:

$F = F^+(\sqrt{-\alpha})$

where:

• $F^+$ is a totally real field (all embeddings land in $\mathbb{R}$)
• $\alpha \in F^+$ is totally positive
• $F$ is totally imaginary (no real embeddings)

2.2 CM Involution

The CM involution is complex conjugation $\sigma: F \to F$, which:

• Fixes the totally real subfield: $\sigma|_{F^+} = \text{id}$
• Acts non-trivially on $F$: $\sigma(\sqrt{-\alpha}) = -\sqrt{-\alpha}$
• Is an automorphism of order 2: $\sigma^2 = \text{id}$

2.3 Embedding Structure

A CM field $F$ of degree $2n$ has:

• $r_1 = 0$ real embeddings
• $r_2 = n$ pairs of complex conjugate embeddings

This means all embeddings land in $\mathbb{C} \setminus \mathbb{R}$, which is crucial for unit norm generation.

2.4 Examples

Field	Degree	Totally Real Subfield	CM Involution
$\mathbb{Q}(i)$	2	$\mathbb{Q}$	Complex conjugation
$\mathbb{Q}(\sqrt{-2})$	2	$\mathbb{Q}$	Complex conjugation
$\mathbb{Q}(\sqrt{-2}, \sqrt{3})$	4	$\mathbb{Q}(\sqrt{3})$	Complex conjugation
$\mathbb{Q}(\sqrt{-2}, \sqrt{3}, \sqrt{5})$	8	$\mathbb{Q}(\sqrt{3}, \sqrt{5})$	Complex conjugation
$\mathbb{Q}(\sqrt{-2}, \sqrt{3}, \sqrt{5}, \sqrt{7})$	16	$\mathbb{Q}(\sqrt{3}, \sqrt{5}, \sqrt{7})$	Complex conjugation

The last field is the H16 base field — see unit-distance-h16-breakthrough.

3. Imaginary Quadratic Fields

3.1 Definition

An imaginary quadratic field is $F = \mathbb{Q}(\sqrt{-D})$ where $D > 0$ is a squarefree positive integer.

• Degree: $[F:\mathbb{Q}] = 2$
• Discriminant: $\Delta_F = -D$ (if $D \equiv 3 \pmod{4}$) or $-4D$ (otherwise)
• Real embeddings: $r_1 = 0$
• Complex embeddings: $r_2 = 1$

3.2 Class Groups

The class group $\text{Cl}(F)$ measures the failure of unique factorization in $\mathcal{O}_F$. Its order $h(F)$ is the class number.

For imaginary quadratic fields:

• $h(F) = 1$ for $D = 1, 2, 3, 7, 11, 19, 43, 67, 163$ (Heegner-Baker-Stark)
• $h(F)$ grows roughly as $\sqrt{D}$ (Gauss class number formula)

3.3 2-Class Rank

The 2-class rank $d$ is the rank of the 2-part of the class group:

$d = \text{rk}_2(\text{Cl}(F))$

By genus theory for imaginary quadratic fields:

$d = \ell - 1$

where $\ell$ is the number of distinct prime factors of $D$. This linear growth is the key limitation that H16 overcomes.

4. Multi-Quadratic Fields

4.1 Construction

A multi-quadratic field is the compositum of $N$ quadratic extensions:

$F = \mathbb{Q}(\sqrt{d_1}, \sqrt{d_2}, \ldots, \sqrt{d_N})$

where $d_i$ are squarefree integers.

4.2 Degree and Galois Group

$[F:\mathbb{Q}] = 2^N$

$\text{Gal}(F/\mathbb{Q}) \cong (\mathbb{Z}/2\mathbb{Z})^N$

Each automorphism independently chooses the sign of each square root.

4.3 2-Class Rank

By genus theory for multi-quadratic fields:

$d = 2^N - 1$

This is the exponential growth that makes multi-quadratic fields superior for the unit distance construction. Compare:

$N$	Degree	$d$	Growth
1	2	1	Linear
2	4	3	Linear
3	8	7	Linear
4	16	15	Linear
5	32	31	Linear

Wait — the growth is $2^N - 1$, which is exponential in $N$. But for fixed $N$, it's linear in the number of quadratic extensions. The key is that $N$ is fixed (e.g., $N = 4$ for H16), and the 2-class rank is determined by $N$, not by the number of ramified primes.

4.4 Discriminant

For a multi-quadratic field ramified at primes $p_1, \ldots, p_\ell$:

$\log H = \frac{1}{2} \sum_{i=1}^{\ell} \log p_i$

This grows linearly with $\ell$, while $d = 2^N - 1$ is independent of $\ell$ (as long as $\ell \ge N$). This is the exponential-vs-linear divergence that powers H16.

5. Complex Multiplication

5.1 Definition

Complex multiplication (CM) refers to endomorphisms of abelian varieties that go beyond the standard multiplication-by-$n$ map. For elliptic curves, CM means the endomorphism ring is larger than $\mathbb{Z}$.

5.2 CM Fields and Class Groups

The CM theory connects:

• The class group of a CM field $F$ to the class group of its totally real subfield $F^+$
• The relative class number $h^-(F) = h(F) / h(F^+)$
• The CM type — a choice of $n$ embeddings out of $2n$ total

5.3 Role in Unit Distance

CM provides:

1. Conjugate prime pairs — complex conjugation pairs prime ideals $(P, \bar{P})$
2. Unit norm generation — Lemma 2.2 uses these pairs to generate elements with $|u| = 1$
3. Global involution — the CM involution is defined globally on $F$, not just locally

6. Class Groups and Genus Theory

6.1 Class Groups

The class group $\text{Cl}(K)$ of a number field $K$ is the group of fractional ideals modulo principal ideals:

$\text{Cl}(K) = I_K / P_K$

where $I_K$ is the group of fractional ideals and $P_K$ is the group of principal fractional ideals.

6.2 Genus Theory

Genus theory computes the 2-class rank of quadratic fields using quadratic reciprocity.

For an imaginary quadratic field $F = \mathbb{Q}(\sqrt{-D})$ with $D = p_1 \cdots p_\ell$:

$d = \text{rk}_2(\text{Cl}(F)) = \ell - 1$

This is because:

• Each prime $p_i$ ramifies in $F$
• The genus characters are $\left(\frac{p_i}{\cdot}\right)$ for $i = 1, \ldots, \ell$
• One relation exists: the product of all genus characters is trivial
• Thus $d = \ell - 1$

6.3 Application to Multi-Quadratic Fields

For multi-quadratic fields, genus theory generalizes:

$d = 2^N - 1$

where $N$ is the number of quadratic extensions. This counts the independent genus characters that are non-trivial.

7. Unramified Extensions

7.1 Definition

An unramified extension $L/K$ is a finite extension where no prime of $K$ ramifies in $L$:

$e(\mathfrak{P}|\mathfrak{p}) = 1 \quad \text{for all primes } \mathfrak{P} \text{ of } L, \mathfrak{p} \text{ of } K$

7.2 Hilbert Class Field

The Hilbert class field $H(K)$ is the maximal unramified abelian extension of $K$. It satisfies:

$\text{Gal}(H(K)/K) \cong \text{Cl}(K)$

7.3 Unramified Pro-p Extensions

The maximal unramified pro-$p$ extension $K^{(p)}$ is the compositum of all unramified extensions of $p$-power degree. Its Galois group is the pro-$p$ completion of $\text{Cl}(K)$.

7.4 The Tower

The unramified pro-$p$ tower is:

$K = K_0 \subset K_1 \subset K_2 \subset \cdots$

where $K_{i+1}$ is the Hilbert class field of $K_i$ (restricted to $p$-power degree).

The Golod-Shafarevich inequality determines whether this tower is infinite.

8. Pro-p Towers and the Frattini Quotient

8.1 Pro-p Groups

A pro-$p$ group is an inverse limit of finite $p$-groups. Key examples:

• $\mathbb{Z}_p$ (p-adic integers)
• Free pro-$p$ groups
• Galois groups of unramified $p$-extensions

8.2 The Frattini Subgroup

For a pro-$p$ group $G$:

$\Phi(G) = G^p [G, G]$

where $G^p$ is the closed subgroup generated by $p$-th powers and $[G, G]$ is the commutator subgroup.

8.3 The Frattini Quotient

$\overline{G} = G / \Phi(G)$

is an elementary abelian $p$-group (vector space over $\mathbb{F}_p$).

• Rank: $d = \dim_{\mathbb{F}_p} \overline{G}$
• Generator rank: $d$ is the minimum number of generators for $G$
• Relation rank: $r$ is the number of independent relations among generators

8.4 Role in the Construction

The Frattini quotient encodes:

1. The 2-class rank $d$ of the base field
2. The capacity for split primes (via the GS inequality)
3. The structure of the unramified tower

9. Lemma 2.2: Unit Norm Generation

9.1 Statement

The critical lemma from unit-distance-remarks.pdf (Page 4):

Lemma 2.2. Let $K$ be a number field embedded in $\mathbb{C}$. Let $k_1, \ldots, k_s$ be positive integers. Let $Q := \prod_{j=1}^s (P_j \overline{P}_j)^{k_j} \subseteq \mathcal{O}_K$ be an ideal. Let $U := \{u \in Q^{-2} : |u| = 1\}$. Then:

$|U| \ge \frac{\prod_{j=1}^s (k_j + 1)}{h(K)}$

9.2 Interpretation

The lemma says:

• Given $s$ conjugate prime pairs $(P_j, \bar{P}_j)$
• With exponents $k_j$ on each pair
• The number of unit-norm elements in the ideal $Q^{-2}$ is at least $\prod (k_j + 1) / h(K)$

9.3 Why CM is Required

The lemma requires:

1. Conjugate pairs $(P_j, \bar{P}_j)$ — provided by complex conjugation
2. Unit norm condition $|u| = 1$ — satisfied because $|P_j| = |\bar{P}_j|$ in the complex embedding
3. CM involution — ensures the norm is balanced across conjugate pairs

Without CM, there are no conjugate pairs, and the lemma cannot be applied.

9.4 The Entropy Formula

For uniform exponents $k_j = k$ and $h(K) \le H$:

$\log |U| \ge s \log(k+1) - \log H$

The entropy per degree is:

$\gamma = \frac{\log |U|}{[K:\mathbb{Q}]} \ge \frac{s \log(k+1) - \log H}{[K:\mathbb{Q}]}$

This entropy drives the exponent $\delta$ in the lower bound.

9.5 Multi-Quadratic Amplification

In a multi-quadratic CM field of degree $2^N$:

• Each rational split prime contributes $2^{N-1}$ conjugate pairs (not just 1)
• The entropy becomes $s \cdot 2^{N-1} \log(k+1) - \log H$
• This is the Galois multiplier that powers H16

10. The Algebraic Toolkit in Action

10.1 The Construction Chain

graph TD
    A["Base Field F<br/>(CM field)"] --> B["Class Group<br/>2-class rank d"]
    B --> C["Golod-Shafarevich<br/>Infinite tower?"]
    C --> D["Unramified Tower<br/>F = F₀ ⊂ F₁ ⊂ F₂ ⊂ ..."]
    D --> E["Split Primes<br/>Killing Frobenius"]
    E --> F["Lemma 2.2<br/>Unit-norm elements"]
    F --> G["Point Configurations<br/>Many unit distances"]
    G --> H["Lower Bound<br/>u(n) = Ω(n^{1+δ})"]

10.2 Parameter Summary for H16

Parameter	Value	Source
Base field $F$	$\mathbb{Q}(\sqrt{-2}, \sqrt{3}, \sqrt{5}, \sqrt{7})$	CM field construction
Degree $[F:\mathbb{Q}]$	16	$2^4$
Totally real subfield $F^+$	$\mathbb{Q}(\sqrt{3}, \sqrt{5}, \sqrt{7})$	CM structure
2-class rank $d$	15	Genus theory: $2^4 - 1$
Unit rank $r_1 + r_2 - 1$	7	$0 + 8 - 1$
Relation rank $r$	22	$d + r_1 + r_2 - 1$
Split primes $t$	17	Legendre symbol sieve
Unramified split primes	$\{59, 131, 251, \ldots, 2411\}$	$(-2/q) = (3/q) = (5/q) = (7/q) = 1$
GS bound $d^2/4$	56.25	$15^2/4$
GS total $r + 2t$	56	$22 + 34$
GS margin	0.25	$56.25 - 56$
Exponent $\delta$	0.019603	Closed-form computation

11. Glossary

Term	Definition
CM field	Totally imaginary quadratic extension of a totally real field
CM involution	Complex conjugation automorphism of a CM field
Class group	Group of fractional ideals modulo principal ideals
2-class rank	Rank of the 2-part of the class group
Genus theory	Method to compute 2-class rank using quadratic reciprocity
Unramified extension	Extension where no prime ramifies
Pro-p tower	Inverse limit of unramified p-power degree extensions
Frattini quotient	Elementary abelian p-quotient of a pro-p group
Relation rank	Dimension of the space of relations among generators
Lemma 2.2	Lower bound on unit-norm elements from conjugate prime pairs
Split prime	Prime that factors completely in the base field

References

• Neukirch, Algebraic Number Theory (1999)
• Washington, Introduction to Cyclotomic Fields (1982)
• Lang, Algebraic Number Theory (1994)
• Cox, Primes of the Form x² + ny² (1989)
• OpenAI, unit-distance-remarks.pdf (Lemma 2.2)
• OpenAI, unit-distance-cot.pdf (Chain of thought)