<s>[INST] Find polynomial over Z2, that has the least degree, for which the field P=GF(64) is the minimum decomposition field. How much such polynomial are? Find polynomial over Z2, that has the least degree, for which the field P=GF(64) is the minimum decomposition field. How much such polynomials are?
Please, answer in detail! I think that these polynomials are all sixth irreducible polynomials, but I don't know how to validate this fact.
 [/INST] This late answer uses computer power to find explicitly the irreductible polynomials of degree six in $\Bbb F_2[X]$.
There exists up to isomorphy only one field with $2^6$ elements, $F=\Bbb F_{64}$, its elements are fixed by the corresponding Frobenius isomorphism, so $F$ contains the roots of
$$x^{2^6}-x\ .$$
sage gives the following factorization for it:
sage: R.<x> = PolynomialRing( GF(2) )
sage: for f, multiplicity in ( x^(2^6) - x ).factor():
....:     print f
....:     
x
x + 1
x^2 + x + 1
x^3 + x + 1
x^3 + x^2 + 1
x^6 + x + 1
x^6 + x^3 + 1
x^6 + x^4 + x^2 + x + 1
x^6 + x^4 + x^3 + x + 1
x^6 + x^5 + 1
x^6 + x^5 + x^2 + x + 1
x^6 + x^5 + x^3 + x^2 + 1
x^6 + x^5 + x^4 + x + 1
x^6 + x^5 + x^4 + x^2 + 1

How many elements in $F$ are roots of a polynomial of degree exactly $6$? There are, by inclusion/exclusion $|F_{2^6}|-|F_{2^3}|-|F_{2^2}|+|F_{2^1}|=2^6-2^3-2^2+2=64-8-4+2=54$ such elements, so we expect $54/6=9$ (edited division...) irreducible polynomials of degree $6$ as factors above.
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