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.