 Binary cyclotomic coset modulo an odd integer

For odd $n$, multiplication by two permutes the residues modulo $n$. Its orbit through $i$ is the binary cyclotomic coset modulo an odd integer. If $\alpha$ has order $n$ in a finite field whose field characteristic is two, the Frobenius automorphism sends ${\alpha }^i$ to ${\alpha }^{2i}$. Hence the minimal polynomial of an algebraic element ${\alpha }^i$ over ${\mathbb{F}}_2$ is ${\prod }_{j\in C_i}(X-{\alpha }^j)$. Its coefficients are fixed by Frobenius and lie in ${\mathbb{F}}_2$, and its degree is the orbit size. Thus a binary cyclic code containing one defining zero of a cyclic code must contain the full orbit as zeros.