<s>[INST] How to find minimal polynomial for finite field 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 $GF(32)$.
 [/INST] 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.
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