<s>[INST] order of the splitting field of $x^5 +x^4 +1 $? what  is the order of the splitting field of 
$x^5 +x^4 +1  = (x^2 +x +1)( x^3 +x+1)$  over $\mathbb{Z_2}$
i thinks  it will  $6$ because  $2.3 = 6$
Pliz help  me...
 [/INST] Yes, it is everything ok. This is an answer, not a comment, in order to insert the following confirmation using computer assistance, here sage:
sage: F = GF(2)
sage: R.<x> = PolynomialRing(F)
sage: f = x^5+x^4+1
sage: f.factor()
(x^2 + x + 1) * (x^3 + x + 1)
sage: K.<a> = f.splitting_field()
sage: K
Finite Field in a of size 2^6
sage:  a.minpoly()
x^6 + x^4 + x^3 + x + 1
sage: f.base_extend(K).factor()
(x + a^3 + a^2 + a)
    * (x + a^3 + a^2 + a + 1)
    * (x + a^4 + a^2 + a + 1)
    * (x + a^5 + a)
    * (x + a^5 + a^4 + a^2 + 1)

The code was minimally rearranged for the last result to fit in the window.
For the first polynomial in the factorization $(x^2 + x + 1) (x^3 + x + 1)$ we need an extension from $\Bbb F_2$ to $\Bbb F_{2^2}$, then from this one a new degree three extension to $\Bbb F_{(2^2)^3}$ to split also the second factor $x^3+x+1$.
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