aspg-40-4465-1792461942 20261020020542 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 2026 6 1 Lee Xu Mathematics Department, University of Chinese Academy of Sciences (CAS), Beijing, China Olalekan Joosati Faculty of Applied Science, Cape Peninsula University of Technology, South Africa We study the reversibility of one-dimensional linear cellular automata (CA) with periodic boundary conditions over a finite field Fq, identifying the global transition map with multiplication by a rule polynomial f (x) in the cyclic group algebra Rn = Fq[x]/(xn−1). We prove that reversibility is exactly equivalent to f (x) being a unit of Rn, and, when gcd(n,q) = 1, we use the classical correspondence between irreducible factors of xn−1 and q-cyclotomic cosets modulo n to derive a closed-form count of the reversible rules of a given neighborhood size: |R×n | = Πi(qdi −1), where the di are the sizes of the q-cyclotomic cosets modulo n. We then resolve the complementary case gcd(n,q)>1: writing n = pam with p = char(Fq) and gcd(m, p) = 1, we show xn−1 = (xm−1)pa , determine the local structure of each factor ring Fq[x]/(g(x)pa) for g irreducible, and obtain the fully general count |R×n| = Πi qdi(pa−1)(qdi −1), valid for every n and every finite field Fq. We give an explicit closed form for the case where m is prime and q is a primitive root modulo m, a density lower bound, an explicit description of the group of reversible rules under composition together with a formula for the exact dynamical period of any reversible rule via its coordinates under the Chinese Remainder Theorem, and a remark on explicit inverse-rule construction. All results are stated and proved in full; numerical instances used to validate the formulas were checked by direct and brute-force computation and are reported without tabulation. 2026 40 44 4465 https://creativecommons.org/licenses/by/4.0/ 10.54216/PMTCS.060106 https://www.americaspg.com/journal/40/article/4465 https://www.americaspg.com/storage/articles/manuscripts/01KYFKMT2D7DSY7EEWK5EQ3330.pdf