Cyclotomic Classes of Order 2

• Cyclotomic Classes of Order 2 are defined as the partition of a finite field's multiplicative group into quadratic residues and nonresidues, foundational for various algebraic and combinatorial applications.
• The framework employs index 2 Gauss sums and residue computations to explicitly construct strongly regular graphs and difference sets with precise eigenvalue properties.
• These classes enable the design of pseudorandom sequences with optimal autocorrelation and support arithmetic factorizations in cyclotomic polynomials for coding theory applications.

Cyclotomic classes of order 2 are fundamental algebraic and combinatorial structures arising from the partition of the multiplicative group of a finite field or residue class ring into cosets of the subgroup of squares. This basic dichotomy—into quadratic residues and nonresidues—underlies numerous constructions and results throughout combinatorics, coding theory, sequence design, and algebraic number theory. The scope of cyclotomic classes of order 2 extends from the explicit construction of strongly regular graphs and difference sets to the fine structure of ideal class groups and the algebraic decomposition of cyclotomic fields.

1. Definitions and Basic Structure

Let $\mathbb{F}_q$ be a finite field with $q$ odd, and $\gamma$ a primitive element of $\mathbb{F}_q^*$. The multiplicative group $\mathbb{F}_q^*$ admits a unique subgroup of index $2$—the subgroup of nonzero squares—so the order 2 cyclotomic classes are: $$C_0 = \{ \gamma^{2k} : 0 \leq k < \tfrac{q-1}{2} \}, \qquad C_1 = \{ \gamma^{2k+1} : 0 \leq k < \tfrac{q-1}{2} \}.$$ $C_0$ consists of the quadratic residues, $C_1$ of the nonresidues. Analogous decompositions arise for cyclic groups like $\mathbb{Z}_n^*$, or for rings $\mathbb{Z}_{2q}$ as in quantum coding applications.

Key properties:

• $C_0$ is a subgroup; $C_1$ is its coset.
• $-1 \in C_0$ if and only if $q \equiv 1 \pmod{4}$.
• For any $a \in \mathbb{F}_q^*$, $a \in C_i$ if and only if the Legendre symbol $(a/q) = (-1)^i$.

This partition generalizes to higher-order cyclotomic classes, but the $N=2$ case (quadratic/residue classes) is the most deeply integrated with both classical and modern algebraic constructions.

2. Graph Constructions and Strong Regularity

Cyclotomic classes of order 2 form the basis of Paley graphs and their generalizations. In the Paley graph $P(q)$, the vertex set is $\mathbb{F}_q$ and two vertices $x$, $y$ are adjacent if and only if $x-y \in C_0$ (1010.4107). This construction is extended by forming Cayley graphs whose connection sets are unions of cyclotomic classes, and by analyzing their spectra via character sums and index 2 Gauss sums.

The index 2 condition—i.e., the characteristic $p$ of $\mathbb{F}_q$ satisfying $\text{ord}_N(p) = \varphi(N)/2$—allows explicit evaluation of relevant Gauss sums and, thus, explicit determination of the graph's eigenvalues. For strongly regularity, it is crucial that the connection set $D$ (possibly a union of cyclotomic classes) is symmetric ($-D = D$) and that the eigenvalue computations reveal exactly two nontrivial restricted eigenvalues (besides the valency).

Such methods produce infinite families of strongly regular Cayley graphs, notably generalizing the classical Paley graphs to new parameter sets determined by these union-of-class constructions and explicit arithmetic constraints on the cyclotomic structure (1010.4107).

3. Cyclotomic Classes of Order 2 in Combinatorial Designs

Skew Hadamard difference sets and related symmetric designs derive from unions of cyclotomic classes of even order, prominently including order 2 (1101.2994). For example, by taking cyclic classes of order $N = 2p_1^m$ and carefully choosing unions $D = \bigcup_{i \in I} C_i$ such that $D \cap (-D) = \varnothing$ and the indices project adequately modulo $p_1^m$, one constructs new difference sets in elementary abelian groups.

Index 2 Gauss sums, rather than cyclotomic numbers, are used to analyze and verify the difference set property by explicit character sum calculations. The method yields numerous new and inequivalent skew Hadamard difference sets and Paley-type partial difference sets, enriching finite geometry and design theory (1101.2994).

4. Explicit Arithmetic in Cyclotomic and Residue Class Rings

Cyclotomic classes of order 2 govern the explicit factorization of certain families of cyclotomic polynomials over finite fields (1011.4857), and underpin the structure of minimal idempotents and arithmetic properties of family algebras in residue class rings like $\mathbb{F}_l[x]/(x^{p^s q^t} - 1)$ (Zhou et al., 31 Jul 2025). In these contexts:

• Polynomials $Q_{2^n r}(x)$ decompose recursively by substituting $x \mapsto x^{2^k}$ in factors of lower order polynomials.
• The splitting $Q_{2^n}(x) = Q_n(-x)$ exploits the order 2 class structure.
• For algebraic decompositions, the cyclotomic cosets parameterize primitive idempotents, whose algebraic interactions—products, squares—are explicitly described using powers and sums weighted by quadratic residue counts and Euler's $\phi$ function.

These results generalize prior work to composite lengths and enable explicit calculation of code parameters and algebraic invariants in coding theory and finite algebra.

5. Sequence Construction, Autocorrelation, and Complexity

Binary and quaternary pseudorandom sequences based on cyclotomic classes of order 2 (or refined generalizations such as Ding–Helleseth classes whose unions recover order 2 structure) exhibit excellent autocorrelation and linear (or 2-adic) complexity, making them robust for cryptographic use (Zhang et al., 2017, Jing et al., 2021).

Construction methodologies involve:

• Assigning sequence values according to cyclotomic residue/nonresidue status in $\mathbb{Z}_n^*$, often controlling signs for zero and other cosets, sometimes via a parameterization $(a, b, c)$ to generate large families.
• Describing autocorrelation via group ring calculations and quadratic Gauss sum analogues.
• Achieving sequences whose algebraic (feedback) complexity attains the maximum possible, i.e., the period, by ensuring that certain greatest common divisors are trivial.

The analytic framework allows a unified treatment across all parameter choices and reveals how algebraic structure from cyclotomic classes informs both randomness and complexity properties.

6. Applications to Number Field Theory and Class Groups

Cyclotomic classes of order 2 appear in number theory in the paper of the $2$-class groups of quadratic fields and the plus/minus decomposition of class groups of cyclotomic fields (1108.6213, Chems-Eddin et al., 2019, Mishra et al., 2020, Prakash, 2022, Chattopadhyay et al., 2023). Highlights include:

• The relations among order 2 ideal classes in quadratic fields are often governed by quartic (i.e., order 4) subextensions of cyclotomic fields, with explicit connections via sums of two squares and genus theory (1108.6213).
• Infinitely many quadratic fields exhibit cyclic $2$-class groups of prescribed order, with Legendre symbol criteria providing the key algebraic test for non-square classes (Dominguez et al., 2012).
• In Iwasawa-theoretic extensions, the stability and rank of the $2$-class group across the cyclotomic $\mathbb{Z}_2$-tower is determined by the decomposition of the base discriminant and explicit congruence and residue symbol conditions, often reflected in the behavior of cyclotomic classes under norm maps and unit indices (Chems-Eddin et al., 2019, Chattopadhyay et al., 2023).

For the maximal real subfield $K^+$ of $K = \mathbb{Q}(\zeta_n)$, the order 2 automorphism given by complex conjugation partitions ideal classes into plus and minus parts, directly relating to cyclotomic classes of order 2 (Mishra, 2020, Mishra et al., 2020, Prakash, 2022). The structure and non-triviality of these subgroups underpin finer questions in class field theory, regularity of primes, and the Cohen–Lenstra heuristics.

7. Broader Implications and Structural Insights

The pervasive appearance of cyclotomic classes of order 2 in combinatorics, number theory, and coding theory reflects a deep connection between binary algebraic partitions and rich combinatorial or arithmetic structures. Key implications include:

• Rigid structural properties in maximum cliques of pseudo-Paley graphs and strongly regular graphs constructed from order 2 classes, including equal-contribution and subfield/subspace patterns (Asgarli et al., 2021).
• The possibility of improving classical combinatorial bounds (e.g., the Delsarte bound on clique numbers) in certain pseudo-Paley graphs where maximum cliques must distribute equally among all cyclotomic classes in the connection set.
• Systematic generalization to higher-order cyclotomic classes and to composite modulus cases, where the binary order 2 framework remains foundational but is enriched by more intricate class decompositions, primitive idempotent structure, and code design (Zhou et al., 31 Jul 2025).

The order 2 case, as the minimal nontrivial division of group structures, thus serves as a canonical starting point and often a facilitating principle in bridging analytic, algebraic, and combinatorial methodologies across a wide spectrum of mathematical research.