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On $p-$ Rationality of Cubic and Quartic Number Fields

This paper establishes a new criterion for determining the $p$ -rationality of complex cubic number fields using third-order recurrence sequences, constructs illustrative examples, and explores the connection between the generalized $abc$-conjecture and $p$ -rationality to identify explicit fields satisfying Greenberg's Generalized Conjecture.

Original authors: Hang Li, Derong Qiu

Published 2026-04-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hang Li, Derong Qiu

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 solve a mystery about the hidden structure of numbers. In the world of mathematics, there are special "neighborhoods" called Number Fields. These aren't places you can visit on a map; they are complex systems built by adding new types of numbers (like roots of equations) to the standard counting numbers.

The paper you're asking about is written by two mathematicians, Hang Li and Derong Qiu. They are investigating a very specific property of these neighborhoods called " $p$ -rationality."

Here is the story of their discovery, explained simply.

1. The Mystery: What is " $p$ -rationality"?

Think of a Number Field as a city. Inside this city, there are "buildings" (numbers) and "roads" connecting them. Sometimes, the city has a hidden flaw: a "torsion" or a "kink" in its structure that makes it messy when you look at it through the lens of a specific prime number, $p$ (like 3, 5, 7, etc.).

The Goal: A city is called " $p$ -rational" if it is perfectly smooth and free of these kinks when viewed through the lens of $p$ .
Why it matters: If a city is $p$ -rational, it behaves very predictably. This is crucial for solving deep, unsolved puzzles in math, like Leopoldt's Conjecture (a rule about how numbers stretch out infinitely) and Greenberg's Generalized Conjecture (a massive theory about how these cities connect to the universe).

For a long time, mathematicians knew how to check if simple cities (like "Quadratic Fields," which are like 2D neighborhoods) were $p$ -rational. But the authors wanted to solve the mystery for more complex, "twisted" cities: Complex Cubic Fields (3D) and Pure Imaginary Quartic Fields (4D). These are much harder to navigate.

2. The New Tool: The "Number Sequence" Test

The authors' biggest breakthrough is a new, simple test to see if these complex cities are $p$ -rational.

Previously, checking this was like trying to inspect every brick in a skyscraper by hand. It was slow and difficult.

The New Analogy:
Imagine you have a special machine that takes a "seed" (a fundamental unit of the city) and grows a sequence of numbers (like a Fibonacci sequence, but more complex).

You feed the seed into the machine.
The machine spits out a long list of numbers: $F_1, F_2, F_3, \dots$
The Test: You look at a specific number in this list (depending on the prime $p$ $p$ you are testing).
- If that number is not divisible by $p^2$ (it doesn't have a "double $p$ " factor), then the city is $p$ -rational! It's smooth.
- If it is divisible by $p^2$ , the city might have a flaw.

This is like checking if a building is stable by tapping a single specific brick. If the brick rings true, the whole building is likely sound. The authors proved that for these complex 3D and 4D cities, this "tapping" method works perfectly.

3. The "ABC" Connection

The paper also connects this to a famous, unsolved problem in math called the Generalized ABC Conjecture.

The Analogy: Imagine the ABC conjecture is a law of physics that says, "You can't have three numbers that are too close to each other without one of them being huge."
The authors show that if this law of physics is true, then there are infinitely many of these complex cities that are $p$ -rational. It's like saying, "If the laws of the universe hold, then there are infinite perfect cities out there waiting to be found."

4. The Evidence: The "Wanted" Lists

To prove their theory, the authors ran computer simulations. They looked at thousands of these complex cities (specifically, those with a "size" or discriminant under 3,000).

Table 1 & 2: These are the "Wanted Lists." They list the specific cities that failed the test (the ones that are not $p$ -rational) for various primes.
The Result: For almost every city they checked, the test worked! They found hundreds of examples of complex 3D and 4D cities that are perfectly $p$ -rational.

5. Why This Matters: The "Greenberg" Prize

The ultimate prize in this game is Greenberg's Generalized Conjecture (GGC).

Think of GGC as a "Grand Unified Theory" for these number cities.
The authors found that if you find a city that is $p$ -rational, you automatically prove that the Grand Unified Theory works for that city.
By finding these new $p$ -rational cities, they have provided concrete proof that the Grand Unified Theory holds in many new, complex situations.

Summary in a Nutshell

The Problem: We didn't know how to easily tell if complex 3D and 4D number systems were "smooth" ( $p$ -rational) or "bumpy."
The Solution: The authors invented a "magic sequence" test. If a specific number in the sequence isn't divisible by $p^2$ , the system is smooth.
The Discovery: They used this test to find hundreds of new smooth systems.
The Impact: This proves that deep mathematical theories (like Greenberg's Conjecture) are true for these complex systems, bringing us closer to understanding the fundamental architecture of numbers.

In short, they built a new, faster "metal detector" that helps mathematicians find the most perfect, well-behaved number worlds in the universe.

1. Problem Statement

The paper addresses the problem of determining $p$ -rationality for number fields that are neither totally real nor quadratic, specifically focusing on complex cubic and pure imaginary quartic number fields.

Definition: A number field $F$ is called $p$ -rational if its Galois group $G_{Sp}(F) = \text{Gal}(M_{Sp}(F)/F)$ (where $M_{Sp}(F)$ is the maximal pro- $p$ extension unramified outside $p$ ) is a free pro- $p$ group.
Context: This property is closely linked to Leopoldt's Conjecture and Greenberg's Generalized Conjecture (GGC).
- $F$ $F$ is $p$ $p$ -rational if and only if:
  1. Leopoldt's conjecture holds for $(F, p)$ .
  2. The maximal $\mathbb{Z}_p$ -torsion submodule of the Galois group of the maximal abelian pro- $p$ extension is trivial (i.e., $T_F = \{1\}$ ).
Open Questions: While significant progress has been made for quadratic fields, the $p$ -rationality of higher-degree fields (cubic and quartic) remains less explored. The authors aim to provide effective criteria to verify this property and explore its connection to the Generalized $abc$-conjecture.

2. Methodology

The authors employ a combination of algebraic number theory, recurrence sequences, and analytic number theory.

A. Criteria for $p$ -Rationality

The paper establishes a refined criterion (Proposition 2.2) for complex cubic and pure imaginary quartic fields. A field $K$ is $p$ -rational if:

The $p$ -Hilbert class field $H_p(K)$ is contained in the composite of all $\mathbb{Z}_p$ -extensions of $K$ ( $\tilde{K}$ ).
There exists a prime ideal $\mathfrak{p}$ of $K$ lying over $p$ such that for a fundamental unit $\varepsilon$ :
$\varepsilon^{p^f - 1} \not\equiv 1 \pmod{\mathfrak{p}^{e+1}}$
where $e$ and $f$ are the ramification index and residue degree of $\mathfrak{p}$ , respectively.

B. Recurrence Sequence Approach (Theorem 1.3)

For complex cubic fields $K = \mathbb{Q}(\varepsilon)$ with fundamental unit $\varepsilon$ having minimal polynomial $f(x) = x^3 - a_2x^2 - a_1x - a_0$ , the authors define a third-order linear recurrence sequence:
$F_{n+3} = a_2 F_{n+2} + a_1 F_{n+1} + a_0 F_n$
with initial conditions $F_0 = F_1 = 0, F_2 = 1$ .
They prove that the non-divisibility of specific terms ( $F_{p-1}, F_{p^2-1}, F_{p^3-1}$ ) by $p^2$ implies the non-congruence condition on the fundamental unit required for $p$ -rationality, depending on the splitting behavior of $p$ in $K$ .

C. Analytic Approach via Generalized $abc$-Conjecture (Theorem 1.4)

Assuming the Generalized $abc$-conjecture, the authors use properties of cyclotomic polynomials and the norm of fundamental units to show that the set of primes $p$ for which $K$ is $p$ -rational has a lower bound of $c \log x$ . This proves the existence of infinitely many such primes.

D. Connection to Greenberg's Generalized Conjecture (GGC)

The paper utilizes the relationship between $p$ -rationality of a CM field and its maximal real subfield. If the maximal real subfield $K^+$ is $p$ -rational and $p \nmid h(K)$ , then GGC holds for the pair $(K, p)$ .

3. Key Contributions

New Computational Criteria: The authors provide explicit, computable criteria involving recurrence sequences to test $p$ -rationality for complex cubic fields. This avoids the need for direct, computationally expensive calculations of class groups in many cases.
Conditional Density Result: Under the Generalized $abc$-conjecture, they prove that for any complex cubic or pure imaginary quartic field $K$ , the number of primes $p \le x$ for which $K$ is $p$ -rational grows at least as $c \log x$ .
GGC Verification: They demonstrate that verifying $p$ -rationality for specific families of fields (e.g., normal closures of complex cubic fields) provides explicit examples satisfying Greenberg's Generalized Conjecture.
Extensive Numerical Verification: The paper includes large-scale computations using PARI/GP and LMFDB data to verify $p$ $p$ -rationality for:
- 419 complex cubic fields with $|d_K| < 3000$ .
- 147 pure imaginary quartic fields with $|d_K| < 3000$ .
- Primes $5 \le p \le 100$ .

4. Main Results

Theorem 1.3: Establishes the link between the recurrence sequence terms $F_n$ and the $p$ -rationality of complex cubic fields. Specifically, if $p \nmid h(K)$ and the sequence term corresponding to the splitting type of $p$ is not divisible by $p^2$ , then $K$ is $p$ -rational.
Theorem 1.4: Proves that assuming the Generalized $abc$-conjecture, the set of $p$ -rational primes for a given complex cubic or pure imaginary quartic field is infinite and has a logarithmic lower bound.
Example 5.1: Shows that for $K = \mathbb{Q}(\sqrt[3]{p^3-1})$ , the field is $p$ -rational for all primes $5 \le p \le 10000$ (except $p=2791$ where $p|h(K)$ , though computer verification suggests it is still $p$ -rational). This leads to the conclusion that the sextic normal closure is also $p$ -rational.
Example 5.6: Constructs infinitely many prime numbers $p$ such that GGC holds for the field $\mathbb{Q}(\sqrt[4]{p^2-4}, \sqrt{-1})$ .

5. Significance

Advancement of GGC: Greenberg's Generalized Conjecture is a major open problem in Iwasawa theory. This paper provides concrete, infinite families of number fields where GGC is verified, moving beyond theoretical existence to explicit construction.
Bridging Conjectures: The work effectively bridges the gap between the Generalized $abc$-conjecture (a deep conjecture in Diophantine analysis) and $p$ -rationality (an algebraic property of Galois groups), showing that the former implies the abundance of the latter.
Algorithmic Efficiency: By reducing the check for $p$ -rationality to the divisibility of terms in a linear recurrence sequence, the authors offer a more efficient computational method for testing these properties in higher-degree fields compared to traditional class group computations.
Data-Driven Insights: The extensive tables (Tables 1 and 2) provide a valuable dataset for the community, identifying specific fields and primes that satisfy or fail $p$ -rationality, and highlighting cases where the class number $h(K)$ is divisible by $p$ (a known obstruction, though not always a definitive one).

In summary, the paper significantly advances the understanding of $p$ -rationality in non-quadratic fields by providing new theoretical criteria, conditional density results, and extensive numerical evidence, thereby generating new explicit examples satisfying Greenberg's Generalized Conjecture.

1. The Mystery: What is "ppp-rationality"?