How to find minimal polynomial if we know some root

Question

So I tried to ask this question about two days ago, but no one could help me and I was wrong with some terminology, and my question was deleted. Im studied abstract algebra and trying to solve some tasks.

So I have an polynomial $X^5+X^4+X^3+X+1$ with coefficients from $F_2[x]$, we know that a - is some root of this polynomial. I have to find minimal polynomial for $a^3+a^2+a$. So I know that $F_2/f(x)$ isomorphic to $F_32$, (as our poly has no roots in $f2$, and deg($f$) = 5)

So if Im not wrong the next step is to search for ratios on a (that is, express the various degrees of a through smaller degrees)(for example we can put a in our poly and try to make something out) and then somehow find this minimal poly

So I have problems with

Am I right?
How to find ratios on a
How it is related to minimal poly for $a^3+a^2+a$

Does this answer your question? How to find minimal polynomial for an element in $\mbox{GF}(2^m)$? + "$f(x)$ has no roots in $F$" does not imply "$f(x)$ is irreducible over $F$"; only the latter is equivalent to "$F[x]/(f)$ is a field". — Amateur_Algebraist, Dec 07 '23 at 15:18
@Amateur_Algebraist Yes f(x) is irreducable over F! Yes the linked question is similar to my, but i have some questiion, In their case he is trying to find minimal poly for B = a^3 If im not wrong i have to take B as a^3+a^2+a? and compute B^0, B^1, B^2......? — Danila Rudnev, Dec 07 '23 at 15:58
Yes, exactly as you say. You'll have to calculate at most the fifth power of $B$, and then construct a nontrivial linear combination of the powers that equals $0$ (say, using Gaussian elimination). Please surround the $\TeX$ formulae with dollar signs (see MathJax tutorial) in your further questions. — Amateur_Algebraist, Dec 07 '23 at 17:39

How to find minimal polynomial if we know some root

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