How to find minimal polynomial for finite field

Question

How can we find minimal polynomials for $\alpha $ in $GF(2^{n})$. What is the general approach to find minimal polynomials. I know about minimal polynomials they are monic etc. In particular i want to know about primitive polynomials of $GF(32)$.

Answer 1

$GF(32)^\times$ is a group of order $31$ and so is cyclic.

You need to factor $x^{31}-1 \bmod 2$ and pick a factor of degree $5$.

$x^5+x^2+1$ is one such factor.

Answer 2

There are $\frac{\phi(2^n-1)}{n}$ primitive polynomials of degree $n$ over $GF(2)$. For $n=5$ we have $30/5=6$ primitive polynomials, namely $$ x^5+x^2+1,\; x^5+x^3+1,\; x^5+x^3+x^2+x+1,\;x^5+x^4+x^2+x+1,\; $$

$$ x^5+x^4+x^3+x+1,\;x^5+x^4+x^3+x^2+1.\; $$ Indeed, the product of all of them, together with the factor $(x+1)$ gives $x^{31}-1$. Using a factorisation algorithm, e.g., the Berlekamp algorithm we obtain the factorization of $x^{31}-1$ over $GF(2)$.

Remark: All irreducible polynomials in $GF(2)[x]$ of degree $2, 3, 5$ are primitive.

References: See here.