Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 12I / c / i / Solution

Since $\alpha$ generates $K^{\times }$, it cannot lie in a proper subfield, so its minimal polynomial over ${\mathbb{F}}_2$ has degree $d$. Write it asThe identity $P(\alpha )=0$ and characteristic two giveMultiplying by ${\alpha }^n$ and applying the linear map $T$ yieldsThis is a binary linear recurrence of order at most $d$, so $(x_n)$ is the output of a linear-feedback shift register of length at most $d$.