The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

Abstract: In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(χ)=\left[G_N(χ^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$ where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $χ$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(χ^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}χ^{ki+kj}(x)e^{2πix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.

• The paper establishes that cyclotomic matrices constructed from Gauss sums are singular if and only if the divisor k is divisible by the prime, revealing a sharp arithmetic criterion.
• It derives an explicit determinant formula that links the matrix determinant to the minimal polynomial constant of Gauss periods, demonstrating integrality and independence from the character generator.
• The study employs matrix decomposition, Galois theory, and combinatorial lemmas to extend classical finite field results to residue rings of prime powers, impacting computational and theoretical number theory.

Cyclotomic Matrices with Gauss Sums over Cyclic Groups

Introduction and Context

The paper "The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups" (2607.02392) investigates the deep arithmetic structure of cyclotomic matrices whose entries are Gauss sums indexed by powers of Dirichlet characters over residue class rings $\mathbb{Z}/N\mathbb{Z}$, with $N$ restricted to odd prime powers. It addresses a nuanced generalization of known results for finite fields, extending the scope from matrices with entries defined through Gauss sums over finite fields to those built over rings, where field techniques are unavailable due to the non-invertibility of zero divisors for $m \geq 2$.

The study of such matrices is motivated by classical results connecting quadratic Gauss sums to determinants of matrices with entries involving roots of unity, and by more recent extensions that have described determinants for matrices formed by Gauss sums over finite fields. The present paper contributes by examining the analog in the context of the cyclic group $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$ and analyzes determinant properties in relation to Gauss periods and the arithmetic of cyclotomic fields.

Mathematical Framework and Main Results

Let $N = p^m$ with $p$ an odd prime, $m \geq 1$, and set $n = \varphi(N) = p^{m-1}(p-1)$. Given a divisor $k$ of $n$, write $N$0. The principal object of study is the $N$1 matrix

$N$2

where $N$3 is a generator of the character group $N$4, and $N$5 denotes the Gauss sum associated to Dirichlet character $N$6 modulo $N$7.

The main theorem establishes two salient facts:

• Singularity Criterion: $N$8 is singular if and only if $N$9.
• Explicit Determinant Formula: If $m \geq 2$0,

$m \geq 2$1

where $m \geq 2$2 is the constant term of the minimal polynomial of the Gauss period $m \geq 2$3, with $m \geq 2$4. Furthermore, the determinant is an integer and independent of the choice of character generator $m \geq 2$5.

These findings significantly generalize prior results for cyclotomic matrices over finite fields, showing that the singularity and determinant criteria are dictated by subtle group-theoretic and field-theoretic properties rather than by explicit evaluations of Gauss sums.

Proof Techniques

The arguments rely critically on multiple layers of structure:

• Matrix Decomposition: $m \geq 2$6 admits a factorization into $m \geq 2$7, where $m \geq 2$8 is a Vandermonde-type matrix with entries $m \geq 2$9 (built from the $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$0-th roots of unity), and $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$1 is diagonal with entries given by Gauss periods evaluated at coset representatives of $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$2.
• Galois Theory: The determinant calculation employs the identification of the minimal polynomial of $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$3 as arising from the action of the Galois group $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$4 and its subgroups corresponding to $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$5.
• Combinatorial Lemmas: The proof leverages a lemma of Newman controlling linear relations among roots of unity, which is vital to establishing the linear independence statements necessary for the determinant's non-triviality.
• Cyclotomic Fields: The explicit connection between the determinant formula and the arithmetic of the intermediate subfields $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$6 of the cyclotomic extension $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$7 elucidates the underlying algebraic structure.

Numerical Significance and Contradictory Claims

A particularly noteworthy and explicit claim is that the determinant is integer-valued, and the formula holds uniformly regardless of the character generator. This is nontrivial since the matrix entries are highly transcendental in nature. Furthermore, the explicit singularity condition—precisely characterized by the divisibility of $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$8 by $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$9—contrasts with the general expectation that such structural properties might only be accessible in finite field contexts.

Implications and Future Directions

Theoretical Implications:

• The correspondence between determinants of character sum matrices and minimal polynomials of Gauss periods demonstrates a profound link between linear algebra over $N = p^m$0, algebraic number theory, and the combinatorial properties of cyclic groups.
• Understanding the singularity loci of such matrices is directly relevant to the arithmetic of cyclotomic extensions, particularly concerning the generation of intermediate fields via Gauss periods.

Practical Implications:

• The precise evaluation of determinants of such structured matrices supports cryptographic applications involving circulant and character matrices over modular rings, especially in schemes leveraging residue rings beyond prime moduli.
• The decomposition may inform algorithms for computing Gauss periods and cyclotomic units in computational number theory and computer algebra systems.

Outlook for Further Research:

Future work may aim to:

• Extend these results to non-prime-power moduli, higher-rank generalizations, or matrices formed from exotic character sums (e.g., higher order Gauss sums, Jacobi sums).
• Explore connections to spectral properties and eigenvalue distributions in the context of random walks on abelian groups and their applications to quantum computation.
• Investigate potential generalizations to cyclotomic matrices associated with more general group schemes, including semidirect products and nonabelian settings.

Conclusion

This study provides a rigorous, explicit determination of the arithmetic and linear-algebraic properties of cyclotomic matrices formed from Gauss sums over residue rings of odd prime powers. The determinant formula ties the analysis directly to the structure of cyclotomic fields and Gauss periods, revealing new algebraic phenomena outside the finite field domain. The singularity characterization, uniformity across character generators, and integrality of determinants offer fertile ground for further investigation into algebraic and computational number theory.