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Let $F=\mathbb{F}_2$ be finite field of order $2$, $f(x)$ be a minimal polynomial of degree $n$ over $F$. Let $K=F(\alpha)$, where $\alpha$ is a root of $f(x)$.

My question is how to deduce the minimal polynomial of $\alpha^i\in K$ over $F$ for any $i$. Are there any computational tools available for this? Please help.

PAMG
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    Are you familiar with cyclotomic cosets (In other words, elementary Galois theory of finite fields)? That gives you a recipe $\prod_j(x-\alpha^{2^ji})$. If not, you can use simple linear algebra like here. In some cases there are tricks. Also, there is the caveat that $\alpha^i$ may generate a proper subfield of $K$, when the minimal polynomial has lower degree. – Jyrki Lahtonen Nov 11 '21 at 19:37
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    Yes I am a bit familiar with that. So, the minimal polynomial is $\prod_{j=0}^{n-1}(x-\alpha^{2^ji})$? Basically, I have placed limits in the recipe provided by you. – PAMG Nov 12 '21 at 09:09
  • Correct. A pitfall is that sometimes there are repetitions among the elements $\alpha^{2^ji}$ and you need to throw those out. For example, if $\alpha$ is a root of the ireducible polynomial $f(x)=x^6+x+1$, then $|K|=64$ implying that $\alpha^{63}=1$. If you want the minimal polynomial of $\alpha^9$, then you need to observe that $$\alpha^{2^3\cdot9}=\alpha^{72}=\alpha^{63+9}=\alpha^{63}\cdot\alpha^9=1.$$ In that case the minimal polynomial is $$(x-\alpha^{9})(x-\alpha^{18})(x-\alpha^{36}),$$ a cubic. – Jyrki Lahtonen Nov 12 '21 at 09:26
  • (cont'd) We don't include the factor $(x-\alpha^{72})$ because it is equal to $(x-\alpha^9)$ and already there. A cubic minimal polynomial is no problem, because the subfield $L=F(\alpha^9)$ is a proper subfield of $K$, and only has $8$ elements. Note that the degrees of the minimal polynomial is a factor of $\deg f(x)=6$ as $3\mid 6$. Degree two also occurs (because $K$ also has a subfield with four elements), check with $\alpha^{21}$. Only when $n$ happens to a prime number, can we deduce that all the minimal polynomials have degree $n$ (save for the minimal polynomials of $0$ and $1$). – Jyrki Lahtonen Nov 12 '21 at 09:30

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