Cyclotomic Matrices

Updated 1 January 2026

Cyclotomic matrices are explicitly defined matrices that encode structural, arithmetic, and spectral properties related to roots of unity and cyclotomic fields.
They are constructed via circulant, Hermitian, or incidence-type frameworks, facilitating applications in primality testing and spectral graph theory.
Their explicit formulations enable deterministic algorithms in number theory and bridge algebraic, combinatorial, and field-theoretic approaches.

A cyclotomic matrix is an explicit matrix encoding structural, arithmetic, or spectral properties closely connected to roots of unity, the theory of cyclotomic fields, cyclotomic polynomials, and their associated Galois actions. The notion of a cyclotomic matrix appears in several mathematical domains, from algebraic number theory and matrix theory to spectral graph theory and arithmetic combinatorics. Constructions range from spectral characterizations of primes via circulant matrices over $\mathbb{Z}$ , to adjacency-type matrices indexed by cyclotomic numbers or character values over finite fields, to Hermitian matrices over cyclotomic integer rings with tightly constrained spectra. These matrices play a pivotal role in bridging algebraic structures—such as cyclotomic field extensions, Galois orbits, or difference sets—with matrix-analytic or graph-theoretic frameworks.

1. Foundational Definitions and Constructions

Cyclotomic matrices in integer algebra and field theory:

A classical model is the integer symmetric matrix $A\in M_n(\mathbb{Z})$ with all eigenvalues in $[-2,2]$ . Via Kronecker’s theorem, the reciprocal polynomial $z^n\chi_A(z+1/z)$ factors into cyclotomic polynomials precisely when $A$ is “cyclotomic” in the sense of McKee–Smyth. In a parallel vein, over $\mathbb{Q}$ and related number fields, cyclotomic matrices frequently arise as circulant matrices constructed from roots of unity. For instance, for each $n\ge1$ , the basic shift matrix $W_n\in \mathbb{Z}^{n\times n}$ is defined by $(W_n)_{i,j}=1$ if $j\equiv i+1\pmod n$ and $0$ otherwise. The composite circulant $C_n=W_n+W_n^2$ features in explicit primality criteria via its minimal polynomial structure (Dinu, 28 Apr 2025).

Cyclotomic matrices over finite fields:

Much recent research examines matrices whose entries are constructed from characters and sums over finite fields. For a prime power $q=p^n$ , and a multiplicative character $\chi$ , the “cyclotomic matrix” often refers to matrices whose entries are values of Jacobi sums $J_q(\chi^i, \chi^j)$ , Gauss sums $G_q(\chi^r)$ , or evaluations of higher-degree characters on structured arguments, e.g., $[\chi^r(s_i+s_j)]$ , with $s_i$ running over $m$ th power residues (Wu et al., 2023, Wu et al., 2024). Other frameworks define cyclotomic matrices via combinatorial incidence constructions, such as cyclotomic numbers of a given order or power-difference distributions (Sun, 17 Nov 2025, Ahmed et al., 2018).

Cyclotomic matrices over rings of integers of number fields:

Classifications of cyclotomic matrices over rings such as the Gaussian integers $\mathbb{Z}[i]$ , Eisenstein integers $\mathbb{Z}[\omega]$ , and general real or imaginary quadratic rings take the Hermitian (or symmetric) matrix $A$ over $R$ with characteristic polynomial in $\mathbb{Z}[x]$ and spectrum constrained to $[-2,2]$ (Greaves, 2011, Greaves, 2012, Taylor, 2010).

2. Spectral and Algebraic Characterization

Eigenvalue and Galois-theoretic structure:

A central organizing principle is that the spectrum of a cyclotomic matrix is deeply entwined with roots of unity and Galois actions. For a circulant $C_n$ , the eigenvalues are $\mu_j = \zeta_n^j + \zeta_n^{2j}$ for $j=0,\ldots,n-1$ ( $\zeta_n$ primitive $n$ -th root of unity). The minimal polynomial $m_{C_n}(x)$ is then formed by collapsing Galois-conjugate eigenvalues, reflecting the symmetries of cyclotomic fields (Dinu, 28 Apr 2025). This mechanism relates matrix factorization properties directly to number-theoretic invariants such as the irreducibility of cyclotomic polynomials, and by extension, to determining primality.

Spectra in combinatorial and finite field settings:

In the finite field context, spectra connect to explicit character sum identities. For matrices $[J_q(\chi^i,\chi^j)]$ , the determinant is computable in closed form: $\det=[J_q(\chi^i,\chi^j)]_{1\le i,j\le q-2}=(q-1)^{q-3}$ (Wu et al., 2023). For quadratic/power-residue matrices or almost circulant structures, eigenvalues are given by sums over cyclic subgroups or moments involving minimal polynomials of certain algebraic integers arising from roots of unity (Wu et al., 2024, Sun, 17 Nov 2025). Over quadratic integer rings, spectral constraints force severe restrictions on admissible edge weights in the associated graphs, with spectra always a subset of $[-2,2]$ (Greaves, 2012, Greaves, 2011, Taylor, 2010).

3. Classification and Structure Theory

Graph-analytic and root-system perspectives:

The classification of cyclotomic matrices over $\mathbb{Z}$ and its (quadratic) extensions is rooted in spectral graph theory and the ADE classification. On $\mathbb{Z}$ , the cyclotomic matrices are precisely adjacency matrices of graphs such that the associated reciprocal polynomial factors into cyclotomic polynomials: the finite/type ADE Dynkin diagrams and their affine extensions, plus several sporadic and tessellation families (0907.0371). The maximality and embeddability results extend to weighted graphs over quadratic integer rings, with each field yielding particular families (e.g., toroidal/cylindrical tessellations, special sporadic graphs with prescribed edge weights) (Greaves, 2012, Greaves, 2011, Taylor, 2010).

Representation-theoretic and algebraic frameworks:

Over a characteristic zero field, any finite extension $\mathbb{K}/\mathbb{F}$ (particularly cyclotomic fields and subfields) admits a representation via a pair of matrices $(A,B)$ over $\mathbb{F}$ such that $\mathbb{K}\cong\mathbb{F}[A]/(B)$ , with $A$ often a circulant or 0–1 companion matrix encoding the algebraic relations of a primitive root of unity (Reddy et al., 2011). The size of such a cyclotomic matrix representation encodes arithmetic complexity: minimal circulant order mirrors conductor degree, while companion matrices can achieve smaller (but bounded below) sizes when the compositional structure is favorable.

Combinatorial and Schur-ring embeddings:

In design theory and the study of power difference sets, cyclotomic numbers and their matrices are best understood through the lens of Schur rings. The associated “cyclotomic matrix” assembles all cyclotomic numbers of a given index, encoding their algebraic relations via ring multiplication and producing matrix product identities, spectral characteristics, and determinant conditions directly interpretable in terms of difference set properties (Sun, 17 Nov 2025).

4. Deteminant Formulas, Explicit Constructions, and Applications

Closed-form determinant evaluations:

A striking feature of cyclotomic matrices in finite field contexts is the existence of explicit determinant formulas, often involving products of classical objects (Jacobi sums, Gauss sums, Beta functions, gamma analogues). Essential results include:

$\det[J_q(\chi^i,\chi^j)]_{1\le i,j\le q-2}=(q-1)^{q-3}$ , congruent mod $p$ to determinantal formulas from Beta integrals and binomial coefficients (Wu et al., 2023).
Determinants of matrices built from Gauss sums yield factorized products depending on the structure of underlying finite fields, with connections to $p$ -adic Gamma functions and minimal polynomials of sums of roots of unity (Wu et al., 2024, Wu et al., 2024).
Explicit evaluations for almost circulant and companion matrix configurations relate determinants to coefficients of polynomials with roots in the relevant cyclotomic extension (Wu et al., 2024, Reddy et al., 2011).
In $\mathbb{Z}$ -graph contexts, determinants of cyclotomic adjacency matrices link to Mahler measure and Lehmer-type problems (0907.0371).

Algorithmic and computational implications:

Cyclotomic matrices provide new deterministic algorithms for number-theoretic problems. Most notably, the irreducibility pattern of the minimal polynomial of $C_n=W_n+W_n^2$ precisely detects the primeness of $n>2$ , yielding a practical primality test with concrete complexity characteristics (competitive with AKS and substantially faster for mid-sized $n$ ) (Dinu, 28 Apr 2025). More generally, the explicit nature of these matrices facilitates the use of linear algebraic tools for testing arithmetic properties, verifying combinatorial configurations (e.g., difference sets), and direct computation of invariants (e.g., class numbers, via determinants associated to Jacobi sum matrices).

5. Interplay with Cyclotomic Fields, Galois Theory, and Number Theory

Matrix avatars of cyclotomic field extensions:

Cyclotomic matrices act as linear-algebraic surrogates for number field extensions. In the circulant context, the polynomial structure of $W_n$ and its combinations encode the Galois theory of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ : orbits under the Galois group correspond to the partitioning of the spectrum, and minimal polynomial irreducibility reflects the arithmetic of field substructures (Dinu, 28 Apr 2025, Reddy et al., 2011). These connections drive a “dictionary” between matrix-theoretic features and field-theoretic invariants (e.g., subfields, factorization patterns, root counts).

Schur rings, difference sets, and algebraic design theory:

The study of power difference sets, cyclotomic numbers, and their related matrices translates geometric and combinatorial properties (existence and structure of difference sets) into conditions on matrix spectra, norms, and determinants (Sun, 17 Nov 2025). The Schur-ring formalism precisely codifies how the algebraic structure of $\mathbb{F}_q$ and its subgroups governs the matrix algebra of cyclotomic matrices.

Connections to Mahler measure and Lehmer’s problem:

In the integer matrix case, cyclotomic matrices demarcate the threshold between matrices with Mahler measure $1$ and those admitting Salem numbers above Lehmer’s constant. The full classification yields a strong version of Lehmer’s conjecture for this matrix class (0907.0371). This further aligns with the role of cyclotomic matrices as the spectrum of “critical” graphs mediating extremal problems in algebraic number theory and dynamical systems.

6. Generalizations, Limitations, and Open Directions

Extensions to higher-degree fields and more general character groups:

Current classification results, though complete for quadratic integer rings and several core settings, remain open for higher-degree extensions of $\mathbb{Q}$ . The rigidity forcing maximal cyclotomic matrices to individual subfields runs into new complexity in mixed or composite extensions (Greaves, 2012). Similarly, determinant and spectrum computations for cyclotomic matrices with higher-power roots or non-cyclic character groups remain difficult, with only conjectural formulas known for some companion matrix configurations (Wu et al., 2023, Wu et al., 2024, Wu et al., 2024).

Combinatorial and spectral rigidity:

Cyclotomic matrices effectively forbid the “mixing” of different irrational weights or non-conjugate roots of unity in the same connected component, as shown by tight degree/spectrum inequalities (Greaves, 2012, Greaves, 2011). For most settings, maximal matrices are unique up to equivalence, and nonintegrality pathologies or spectral outliers do not persist beyond small dimensions.

Potential for new primality and factoring algorithms:

The cyclotomic–circulant approach to primality testing offers a deterministic, algebraically interpretable alternative to probabilistic and classical deterministic protocols (Dinu, 28 Apr 2025). The method both prompts further analysis of cyclotomic matrix spectra for generalized circulant types and suggests hybrid approaches combining field-theoretic and matrix-analytic insights for more general factorization or arithmetic property testing.

References (arXiv ID):