Vtyjkyuk

what is the order of the splitting field of

$x^5 +x^4 +1 = (x^2 +x +1)( x^3 +x+1)$ over $mathbbZ_2$

i thinks it will $6$ because $2.3 = 6$

Pliz help me...

Because $x^5 +x^4 +1 = (x^2 +x +1)( x^3 +x+1)$ and the factors are irreducible, the splitting field of $x^5 +x^4 +1$ has degree at least $2 cdot 3 = 6$.

On the other hand, the splitting field of $x^5 +x^4 +1$ is contained in $mathbb F_2^6$, an extension of degree $6$. Hence this is the splitting field. Indeed, WA tells us that
$$
x^2^6-x = x (x + 1) (x^2 + x + 1) (x^3 + x + 1) cdots
$$

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 $x^3+x+1$.