Galois Group of a Polynomial Problem help

Question

Can someone help me find the Galois group of the following polynomial:

$f = x^5 + x^4 + x^3 + x, f \in \Bbb Z_2[X]$

Don't know how to use LaTeX here, so forgive me about that.

Answer 1

You can factor that polynomial as

$$f(x) = x(1 + x^2 + x^3 + x ^4) = x(x+1)(1+x+x^3)$$

$(1+x+x^3)$ is irreducible because it has no roots (and if it was reducible one of its factors would have degree $1$).

Let $\alpha$ be a root of $1 + x + x^3$. Then, $[\mathbb{F}_2 (\alpha): \mathbb{F}_2] = 3$. Let $L$ be the splitting field of $f$, which is also the splitting of $1 + x + x^3$.

Now observe the following:

$$(\alpha^2)^3 + \alpha^2 + 1 = (1 + \alpha + \alpha^3)^2 = 0$$

Since $1 + x + x^3$ is the minimal polynomial of $\alpha$; $\alpha^2 \neq \alpha$. Hence, we get $2$ roots of $1 + x + x^3$ in $\mathbb{F}_2 (\alpha)$, so all three of them must be there.

The Galois group has the same order as the degree of the splitting field, so it has order $3$, which means it is $C_3$.