Cyclic Algebras: Structure & Applications

• Cyclic algebras are central simple algebras defined by cyclic Galois extensions and explicit generator relations, serving as foundational elements in Brauer group theory.
• They admit multiple presentations via crossed products, symbol algebras, and Witt vector constructions, enhancing both computational and theoretical approaches.
• Their stability under tensor products and crucial role in division algebra theory demonstrate significant applications in algebra, topology, and quantum field theory.

A cyclic algebra is a central simple algebra constructed from a cyclic Galois extension, playing a pivotal role in the structure theory of central simple algebras and the explicit description of the Brauer group. Cyclic algebras admit multiple explicit presentations—via crossed products using group cohomology, via symbol algebras with explicit generators and relations, and via Witt vector constructions in characteristic $p$—and serve as the foundational building blocks for $p$-torsion in the Brauer group. They are characterized by the existence of a cyclic maximal subfield, have a canonical norm structure, and satisfy strong generation properties with respect to algebraic $K$-theory and cohomological invariants.

1. Structure and Presentations

Let $F$ be a field and $n$ a positive integer. A central simple algebra $A/F$ of degree $n$ is called a cyclic algebra if it admits a presentation:

• A cyclic Galois extension $L/F$ of degree $n$ with Galois group $G=\langle \sigma \rangle$.
• An element $a\in F^\times$.

The algebra $A = (L/F, \sigma, a)$ is defined as the $F$-vector space $\bigoplus_{i=0}^{n-1} L x^i$ with multiplication determined by $x^n=a$ and $x \ell = \sigma(\ell) x$ for all $\ell \in L$. If $F$ contains a primitive $n$th root of unity $\zeta$, an alternative symbol algebra presentation is

$$A = F\langle x, y \mid x^n = \alpha,\, y^n = \beta,\, yx = \zeta x y \rangle$$

for $\alpha, \beta \in F^\times$, denoted $A = (\alpha, \beta)_{n,F}$ (Chapman, 2014).

In characteristic $p$, nontrivial cyclic algebras arise via Artin–Schreier–Witt theory, with the shorter notation $[\omega,\beta)_{p^m,F}$; the defining generators and relations employ Witt vectors and higher degree symbol manipulations (Chapman, 21 Oct 2025).

2. Cohomological and K-Theoretic Aspects

Cyclic algebras are classified by Galois cohomology: $$\operatorname{Br}(F)[n] \cong H^2(\operatorname{Gal}(\overline{F}/F), \mu_n)$$ The class of $(L/F, \sigma, a)$ corresponds to the cup product $(L)\cup(a)$ via the Kummer isomorphism $F^\times/(F^\times)^n \cong H^1(F, \mu_n)$. All symbol algebras $(\alpha,\beta)_{n,F}$ are cyclic and their products and relations encode the structure of the $n$-torsion in $\operatorname{Br}(F)$ (Chapman, 2014).

For $p$-torsion and in the presence of roots of unity, cyclic algebras are known to generate the entire torsion part of $\operatorname{Br}(F)$ under tensor product and suitable norm relations. In characteristic $p$, the Artin–Schreier–Witt symbol realizes every $p^m$-torsion class as cyclic.

3. Tensor Products and Cyclicity Results

A fundamental property of cyclic algebras is their stability under tensor products, as encapsulated by Albert's theorem. In characteristic $p>0$, Chapman provides an explicit algorithmic determination of the resulting cyclic class via Witt-vector symbols: $$[\omega, \beta)_{p,F} \otimes [\alpha, \gamma)_{p,F} \simeq [(\delta, \tau),\; \delta\gamma^p)_{p^2,F}$$ with $\delta = \beta + (\alpha-\beta)^p$ and $\tau = \omega \delta \beta^{-1}$, and similar induction for higher degrees (Chapman, 21 Oct 2025). This confirms that the tensor product of two cyclic $p$-algebras of degree $p$ is again cyclic of degree $p^2$, and—by induction—arbitrary tensor products of cyclic algebras generate all $p$-power torsion in $\operatorname{Br}(F)$.

4. Cyclic Algebras in the Theory of Division Algebras

Every division algebra $D$ of degree $n$ that is split by a cyclic Galois extension of degree $n$ is cyclic. Rowen–Saltman and Matzri refined these cyclicity criteria for algebras split by dihedral extensions $E/F$ of degree $2n$ with $n$ odd; for $n=5$, no assumption on $F$ containing a primitive 5th root of unity is required. If $D$ is central simple of degree $5$ split by a $D_5$-extension, then $D$ is cyclic; consequently, the 5-torsion in $\operatorname{Br}(F)$ is generated by cyclic algebras (Matzri, 2014).

The proof strategy involves:

• Reduction to the cyclic half of the extension.
• Element construction with vanishingly many characteristic polynomial coefficients.
• Application of Coray’s descent lemma for cubic forms to descend zeros from a quadratic extension to $F$.
• Application of Albert's cyclicity criterion for establishing cyclic structure (Matzri, 2014).

5. Kummer Subspaces and Maximal Subspaces

Cyclic algebras contain important families of "Kummer subspaces," i.e., $F$-vector subspaces consisting of elements $a$ with $a^n\in F$, $a^k \notin F$ for $1\le k<n$. Maximal Kummer subspaces in tensor products of cyclic algebras are classified: for $A=\bigotimes_{k=1}^n (\alpha_k,\beta_k)_{p,F}$, there are explicit inductive constructions $V_k$ of dimension $pk+1$, shown to be maximal. For degree-3, a complete graph-theoretic classification is obtained, with maximal monomial Kummer spaces corresponding to special tournaments, and the maximal dimension in the generic case is $3n+1$ (Chapman, 2014).

These Kummer subspaces control the explicit description of torsion elements in the Brauer group and constrain structures such as sub-Severi–Brauer varieties.

6. Cyclic Algebras in Categorical and Topological Frameworks

In higher categorical settings, cyclic algebras extend beyond central simple algebras to cyclic algebras over operads in symmetric monoidal bicategories. A cyclic algebra over, for example, the associative or the framed little 2-disks operad, is encoded as an object with a nondegenerate symmetric pairing and a compatible structure up to coherent isomorphism. Explicit equivalences relate:

• Cyclic associative algebras to pivotal Grothendieck–Verdier categories.
• Cyclic framed little disks algebras to ribbon Grothendieck–Verdier categories (Müller et al., 2020).

These categories play a role in mapping class group representations, modular functors, and quantum topology. For instance, any ribbon Grothendieck–Verdier category produces a system of handlebody group representations, generalizing classical modular category constructions.

7. Applications and Broader Significance

Cyclic algebras have profound impact in the structure theory of the Brauer group, Galois cohomology, explicit construction of division algebras, norm forms, and norm equations. Computationally explicit presentations—especially the symbol algebra and Witt vector symbol formulas—are crucial for algorithmic approaches to explicit line bundle and division algebra constructions. In topological field theories, modular categories arising from cyclic categorical algebras underpin the construction of quantum invariants and mapping class group actions (Müller et al., 2020). The explicit generation of torsion in the Brauer group by cyclic algebras substantiates the Merkurjev–Suslin philosophy that $p$-torsion should be generated in this way, providing a foundational tool for research in both the algebraic and arithmetic theory of central simple algebras.

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Key References:

• Matzri, "All dihedral division algebras of degree five are cyclic" (Matzri, 2014)
• Chapman, "Kummer Subspaces of Tensor Products of Cyclic Algebras" (Chapman, 2014)
• Chapman, "The Cyclicity of Tensor Products of Cyclic $p$-Algebras" (Chapman, 21 Oct 2025)
• Müller–Woike, "Cyclic framed little disks algebras, Grothendieck-Verdier duality..." (Müller et al., 2020)