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On the fourth power level of $\mathfrak{p}$ -adic completions of biquadratic number fields

This paper computes the fourth level of $\mathfrak{p}$ -adic completions for biquadratic number fields under specific ramification conditions, a result that facilitates determining the fourth level for any quartic number field and yields lower bounds for biquadratic fields.

Original authors: Kazimierz Chomicz

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kazimierz Chomicz

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 a world where numbers aren't just tools for counting your allowance or calculating a pizza order, but characters in a vast, invisible city. In this city, called "Number Theory," mathematicians are like urban planners trying to understand how different neighborhoods (fields) connect. One of the most interesting jobs they have is figuring out how to build "sums of powers." Think of a "power" like a number multiplied by itself a few times (like $2 \times 2 \times 2 \times 2$ ). The big question is: if you want to build a specific target number (like -1) by adding up these powers, how many of them do you need?

Sometimes, you can do it with just one or two. Other times, you might need a whole bucketful. And in some strange neighborhoods, you might never be able to build that target at all, no matter how many you throw in. Mathematicians call the smallest number of pieces you need the "level" of that neighborhood. If you need 15 pieces, the level is 15. If you can't do it, the level is infinity. This paper focuses on a specific, tricky version of this puzzle: the "fourth level," which asks how many fourth powers you need to add up to get -1. Why does anyone care? Because these levels are like the DNA of a number field; knowing them helps us understand the hidden structure of numbers, which is crucial for everything from cryptography to pure math logic.

Now, enter Kazimierz Chomicz, a high school student with a very sharp mind, who decided to tackle a particularly stubborn part of this puzzle. He looked at a special type of number city called a "biquadratic number field." You can think of these as cities built by stacking two square-root neighborhoods on top of each other (like $\sqrt{m}$ and $\sqrt{n}$ ). The paper's main job was to figure out the "fourth level" for these cities, but specifically by looking at their "p-adic completions." If a number field is a city, a p-adic completion is like zooming in on a specific street corner (a prime ideal) to see the microscopic details of how numbers behave right there.

Chomicz's paper is a masterclass in zooming in. He focused on the most difficult street corner possible: the one where the "ramification index" is 4. In our city analogy, this is a corner where the streets are so twisted and tangled that the usual rules of thumb don't work. For most other corners, mathematicians already knew the answer. But for this specific, messy corner (where the ramification index is 4 and the "inertial degree" is 1), the answer was a mystery. Chomicz spent the paper doing a massive amount of detailed arithmetic, essentially checking every possible combination of numbers to see how they add up. He didn't just guess; he proved exactly what the answer is based on specific conditions.

Here is what he found: The answer isn't just one single number. It depends on a secret code hidden in the numbers $m$ and $n$ that define the city. Depending on how these numbers behave when divided by 4, 8, or 16, the fourth level can be 1, 2, 3, 4, 6, or even 15. He created a detailed map (Theorem 3.1) that tells you exactly which number you get for any biquadratic field. For example, if your numbers leave a remainder of 3 when divided by 4, the answer might be 1, 2, or 3 depending on even finer details. If they leave a remainder of 1, the answer could be 15 or 2.

One of the most exciting moments in the paper is a specific example: the field $\mathbb{Q}(\sqrt{-2}, \sqrt{-6})$ . Chomicz showed that for this city, the fourth level is exactly 3. This is a big deal because, until now, 3 was a number that mathematicians weren't sure could even exist as a fourth level for any number field. He didn't just calculate a lower bound; he found an explicit formula showing that three specific fourth powers add up to zero, proving the level is indeed 3.

The paper concludes with a bold guess (a conjecture). Chomicz suggests that for any number field, the true fourth level is simply the highest "level" found at any of its street corners (p-adic completions). If this guess is true, it means that instead of trying to solve the whole city at once, mathematicians could just zoom in on the corners, solve the puzzle there, and take the biggest answer they find. It turns a potentially impossible task into a manageable one. While this specific result is a proven fact for the cases he checked, the final guess about all number fields remains a "conjecture"—a strong hunch that needs more testing. But for the specific biquadratic fields he studied, the mystery of the fourth level has been largely solved, revealing a beautiful, conditional pattern hidden inside the numbers.

Technical Summary: On the Fourth Power Level of p-adic Completions of Biquadratic Number Fields

Problem Statement
The paper addresses the computation of the fourth level, denoted $s_4(K)$ , of a number field $K$ . The fourth level is defined as the smallest positive integer $g$ such that the equation $x_1^4 + x_2^4 + \dots + x_g^4 = -1$ is solvable in $K$ . If no such integer exists, $s_4(K) = \infty$ . The author notes that determining bounds for $s_4(K)$ is a technically difficult problem, comparable to computing Waring numbers (Pythagoras numbers) for fields.

The specific focus is on quartic number fields (degree 4). A fundamental lower bound for $s_4(K)$ is derived from its $p$ -adic completions $K_p$ at prime ideals $p$ dividing 2, via the inequality $s_4(K) \geq s_4(K_p)$ . Previous work (cited as [17]) had computed $s_4(K_p)$ for all cases of ramification index $e$ and inertial degree $f$ except for the specific case where $e=4$ and $f=1$ . This paper aims to fill that gap and apply the result to biquadratic number fields of the form $K = \mathbb{Q}(\sqrt{m}, \sqrt{n})$ .

Methodology
The methodology relies on the properties of $p$ -adic valuations and Hensel's Lemma.

Reduction to Finite Rings: For a prime ideal $p | (2)$ with ramification index $e$ and inertial degree $f$ , the computation of $s_4(K_p)$ is reduced to finding the smallest number of fourth powers summing to zero modulo $p^{3e+1}$ (or $p^{4e+1}$ ). In the specific case of $e=4, f=1$ , this requires analyzing sums modulo $p^{13}$ .
Case Analysis via Valuation: The author introduces an element $\tau \in \mathfrak{p} \setminus \mathfrak{p}^2$ and analyzes the valuation of the expression $\tau^4 + 2$ . The value of $s_4(K_p)$ depends on the highest power of $\mathfrak{p}$ dividing $\tau^4 + 2$ . The paper systematically examines cases where $p^\alpha \parallel (\tau^4 + 2)$ for $\alpha$ ranging from 5 to 13.
Algebraic Computation: The proofs involve extensive algebraic manipulation of fourth powers of elements in the residue ring $\mathcal{O}_K/\mathfrak{p}^{13}$ . The author defines "good sums" (sums with an even, non-zero number of terms not divisible by $p$ ) and computes their remainders.
Application to Biquadratic Fields: To apply the general theorem to biquadratic fields $K = \mathbb{Q}(\sqrt{m}, \sqrt{n})$ , the paper determines the factorization of the ideal $(2)$ in the ring of integers $\mathcal{O}_K$ . Since these fields are not always monogenic, the standard Kummer-Dedekind theorem is not directly applicable. Instead, the author constructs the ring of integers as a quotient of a polynomial ring over $\mathbb{Z}$ and reduces it modulo 2 to analyze the prime ideal structure using computational algebra (Macaulay2).

Key Contributions and Results

Theorem 2.5 (Main General Result): The paper provides a complete classification of $s_4(K_p)$ for any number field $K$ with a prime ideal $p | (2)$ where $e(p)=4$ and $f(p)=1$ . The value of $s$ is shown to be in the set $\{1, 2, 3, 4, 5, 6\}$ . The specific value is determined by the valuation of $\tau^4 + 2$ and the divisibility of specific polynomial expressions in $\tau$ by powers of $p$ (up to $p^{13}$ ). This generalizes previous results from [17] which covered all other combinations of $e$ and $f$ for quartic fields.
Theorem 3.1 (Application to Biquadratic Fields): The paper computes the lower bounds for $s_4(K)$ $s_{4} (K)$ for biquadratic fields $K = \mathbb{Q}(\sqrt{m}, \sqrt{n})$ $K = Q (m, n)$ based on the congruence classes of $m, n$ $m, n$ modulo 4 and 8. The results categorize $s_4(K_p)$ $s_{4} (K_{p})$ into values $\{1, 2, 3, 4, 6, 15\}$ ${1, 2, 3, 4, 6, 15}$ depending on the factorization of $(2)$ $(2)$ and the specific congruences of the field parameters.
- For instance, if $m \equiv n \equiv k \equiv 1 \pmod 8$ (where $k = mn/\gcd(m,n)^2$ ), then $s_4(K_p) = 15$ .
- If $m \equiv 3, n \equiv k \equiv 2 \pmod 4$ , the value depends on the divisibility of $n+k$ by powers of 2 and the residue of $nk/4 \pmod{16}$ .
Explicit Example: The paper notes that for $K = \mathbb{Q}(\sqrt{-2}, \sqrt{-6})$ , the lower bound is 3, and explicitly demonstrates that $s_4(K) = 3$ via a specific sum of four fourth powers equaling zero.

Significance and Claims
The paper claims to provide the first explicit computation of the fourth level for the previously unresolved case of $e=4, f=1$ in quartic number fields. By combining this new theorem with existing results for other $e, f$ pairs, the author asserts that the fourth level of the $p$ -adic completion is now computable for every possible case of a quartic number field, provided the integral basis is known.

The author highlights that in all cases where both the lower bound (from $p$ -adic completions) and the actual field level could be computed, they were found to be equal. Based on this observation, the paper proposes Conjecture 4.2: For any number field $K$ with a finite fourth level, $s_4(K) = \max_{p|(2)} s_4(K_p)$ . If true, this would reduce the computation of the fourth level of a number field to a finite task involving the factorization of the ideal $(2)$ .

The paper concludes by noting that $s_4(\mathbb{Q}(\sqrt{-2}, \sqrt{-6})) = 3$ appears to be the first known number field with a fourth level of 3, and poses the open problem of determining all possible values that $s_4(K)$ can attain for number fields.

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