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The classification of real quadratic fields which satisfy Hammarhjelm's condition

This paper proves that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition—those with discriminants 8, 5, 13, 29, 53, 173, and 293—by demonstrating that their fundamental units are small relative to the discriminant and utilizing genus theory alongside Biro's classification of class number one fields.

Original authors: Zeev Rudnick

Published 2026-07-01

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Zeev Rudnick

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to find a very specific, rare type of lock. In the world of mathematics, these "locks" are called real quadratic fields. They are special number systems built by adding the square root of a whole number (like $\sqrt{2}$ or $\sqrt{5}$ ) to the regular counting numbers.

Most of these number systems are messy. If you try to break numbers down into their basic building blocks (prime factors), you might find that the same number can be broken down in different ways, like a puzzle with missing pieces. However, some of these fields are perfectly tidy: they have unique factorization, meaning every number breaks down into primes in exactly one way, just like our regular counting numbers.

The Special Rule: Hammarhjelm's Condition

The paper focuses on a tiny, elite club of these tidy number systems. To join the club, a field must satisfy two rules:

The Tidy Rule: It must have unique factorization (no messy puzzles).
The "Empty Room" Rule (Condition H): This is the tricky part. Imagine the number system as a grid of points in a 2D room. There is a special "key" number in this system called the fundamental unit (think of it as the master key that unlocks the whole system).

The rule says: If you draw a specific rectangular box in this room—defined by the size of that master key—there must be no other points inside that box. It must be completely empty.

Analogy: Imagine a giant, empty parking lot. The "master key" is a giant truck parked at one end. The rule says that if you draw a specific zone right next to that truck, no other cars (numbers) are allowed to park there. If even one tiny car sneaks in, the field is disqualified.

The Big Discovery

For a long time, mathematicians knew of five fields that passed this test (with numbers $D = 2, 5, 13, 29, 53$ ). The author of this paper, Ze'ev Rudnick, went on a hunt to see if there were any others hiding in the shadows.

The Result: He proved that there are exactly seven such fields in the entire universe of mathematics. No more, no less.

The complete list of these seven "perfect" fields corresponds to the numbers:
2, 5, 13, 29, 53, 173, and 293.

The first five were known; the paper adds 173 and 293 to the list and proves that no other numbers can ever join.

How the Detective Solved the Case

The paper uses a few clever tricks to narrow down the search:

The "Small Key" Clue: The author realized that for a field to pass the "Empty Room" test, its master key (the fundamental unit) has to be surprisingly small compared to the size of the field. If the key is too big, the "empty zone" becomes huge, and it's statistically impossible for it to remain empty. This clue immediately ruled out almost all huge numbers.
The "Family Tree" Strategy: The author grouped the remaining candidates into families.
- One family is called Yokoi's family. These are fields where the number $D$ looks like a square number plus 4 (e.g., $3^2 + 4 = 13$ ). A previous mathematician had already listed the "tidy" members of this family. The author checked them against the "Empty Room" rule and found that the new candidates (173 and 293) fit perfectly.
- Another family is called Richaud-Degert. The author proved that these fields always fail the "Empty Room" test because they always have a "car" parked in the forbidden zone.
- Other families were ruled out using logic about how prime numbers multiply together (Genus Theory).

Why Does This Matter?

The paper mentions that these specific fields have recently appeared in the study of "visible points" in complex geometric patterns (called cut-and-project sets). Think of these patterns as a way to create beautiful, non-repeating tilings (like a Penrose tiling) or crystal structures.

The "Empty Room" condition ensures that when you look at these patterns, you can see certain points clearly without them being blocked by others. The author's work confirms that there are only seven specific types of number systems that create these perfect, unobstructed views.

In summary: The paper is a mathematical census. It proves that out of the infinite number of possible quadratic fields, only seven satisfy a strict geometric "no trespassing" rule, and it provides the exact list of those seven.

Technical Summary: The Classification of Real Quadratic Fields Satisfying Hammarhjelm's Condition

Problem Statement
This paper addresses the classification of real quadratic fields $K = \mathbb{Q}(\sqrt{D})$ (where $D > 1$ is squarefree) that satisfy "Hammarhjelm's condition." A field satisfies this condition if two criteria are met:

The ring of integers $\mathcal{O}_K$ has unique factorization (i.e., the class number $h_K = 1$ ).
The Minkowski lattice $f(\mathcal{O}_K) \subset \mathbb{R}^2$ , defined by the embedding $\alpha \mapsto (\alpha, \sigma(\alpha))$ , contains no points within the open rectangle $(1, \lambda) \times (-1, 1)$ , where $\lambda > 1$ is the fundamental unit of $K$ .

These fields have recently appeared in the study of visible points in algebraic cut-and-project sets. Prior to this work, the known examples were limited to $D \in \{2, 5, 13, 29, 53\}$ . The objective is to determine the complete list of such fields.

Methodology
The proof proceeds by establishing a relationship between Hammarhjelm's condition and the size of the fundamental unit relative to the discriminant, followed by a case-by-case analysis using genus theory and existing classifications of class number one fields.

Bounding the Fundamental Unit:
The author first demonstrates that Hammarhjelm's condition forces the fundamental unit $\lambda$ to be small relative to the discriminant $\Delta_K$ .
- If $D \equiv 2, 3 \pmod 4$ , then $\lambda < 2\sqrt{D} = \sqrt{\Delta_K}$ .
- If $D \equiv 1 \pmod 4$ , then $\lambda < \frac{\sqrt{D}+1}{2} = \frac{\sqrt{\Delta_K}+1}{2}$ .
  This bound is derived by constructing specific elements in $\mathcal{O}_K$ that would fall into the forbidden rectangle if $\lambda$ were larger.
Genus Theory and Narrow Class Groups:
The paper utilizes genus theory to restrict the possible discriminants for fields with class number one. Specifically, the narrow class number $h_K^+$ is divisible by $2^{r(K)-1}$ , where $r(K)$ is the number of distinct prime factors of $\Delta_K$ . For $h_K=1$ , this implies $r(K) \le 2$ .
- The case $r(K)=2$ with $D \equiv 1 \pmod 4$ (where $\Delta_K = pq$ ) is analyzed using the 4-rank of the narrow class group (via Rédei matrices). It is shown that if $p, q \equiv 1 \pmod 4$ and $q = p+4$ , the 4-rank is 1, implying $h_K > 1$ .
Classification by Discriminant Type:
The proof divides the problem based on the congruence of $D$ modulo 4:
- $D \equiv 2, 3 \pmod 4$ : The author shows that no such fields satisfy the condition for $D > 2$ . For $D=2$ , the condition holds. For $D \equiv 3 \pmod 4$ , the absence of units with negative norm and the bound on $\lambda$ lead to a contradiction for any prime $D$ .
- $D \equiv 1 \pmod 4$ :
  - Prime Discriminant ( $D=p$ ): The bound on $\lambda$ forces the fundamental unit to have norm $-1$, placing $D$ in the form $D = (2m+1)^2 + 4$ . This identifies the fields as a subset of Yokoi's family.
  - Composite Discriminant ($D=pq$): The analysis rules out cases where $p, q \equiv 1 \pmod 4$ (due to the 4-rank argument) and cases where $p, q \equiv 3 \pmod 4$ (which fall into the "narrow Richaud-Degert" family, shown to fail Condition H for $D > 5$ ).
Application of Biró's Classification:
For the fields identified as belonging to Yokoi's family ( $D = m^2 + 4$ ), the paper relies on Biró's (2003) classification of class number one fields within this family. Biró determined that the only squarefree discriminants of this form with class number one are $D \in \{5, 13, 29, 53, 173, 293\}$ .

Key Contributions and Results
The main result is Theorem 1.1, which states that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition. These correspond to the discriminants:
$D \in \{2, 5, 13, 29, 53, 173, 293\}$

The paper provides a rigorous proof that:

The previously known examples ( $D=2, 5, 13, 29, 53$ ) are part of a complete list.
Two additional fields, $D=173$ and $D=293$ , satisfy the condition.
No other real quadratic fields satisfy the condition.

The proof establishes that Hammarhjelm's condition implies the fundamental unit is small relative to the discriminant, which restricts the search space to specific families of fields (Yokoi's family and narrow Richaud-Degert extensions) where class number one fields are already classified.

Significance
The paper resolves the problem of determining all real quadratic fields satisfying Hammarhjelm's condition, a problem motivated by recent developments in the study of visible points in algebraic cut-and-project sets. By combining elementary bounds on the fundamental unit with deep results from genus theory and the classification of Yokoi's family, the author provides a definitive and finite list of such fields, extending the known set from five to seven examples.

The Special Rule: Hammarhjelm's Condition

The Big Discovery

How the Detective Solved the Case

Why Does This Matter?

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