Multiplication Orbits and Finite-Field Decompositions

We study the orbits of multiplication by T on R_n = F₂[T]/(Tⁿ + T + 1) for every integer n ≥ 2. Let V denote the root space of L(Y) = Y^(2ⁿ) + Y² + Y over an algebraic closure of F₂. We construct an explicit F₂-linear isomorphism between R_n and V that intertwines multiplication by T with the Frobenius operator. This correspondence identifies each multiplication orbit with an irreducible factor of L, whose degree equals the orbit length. Using these factors, we obtain the finite-field decomposition of F₂[Y]/(L(Y)) as a product of finite fields, where the degrees of the field factors correspond to the multiplication-orbit lengths. This is the decomposition formulated in Conjecture 3.8 of Laurent Bartholdi's 2000 paper, Lamps, Factorizations and Finite Fields.