= Binary cyclotomic coset modulo an odd integer
{title2=$C_i=\{i,2i,4i,\ldots\}\pmod n$}
= Binary cyclotomic cosets modulo an odd integer
{synonym}
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.
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