show that $x^2+x+1 \in F[x]$ is irreducible

Question

So this is in regards to exam revision. The complete question is to construct a field of order 32, then show that $x^2+x+1\in F[x]$ is irreducible.

So I began by constructing the field

$$\begin{align} F[x]=&Z_{2}[x]/(x^5+x^2+1)\\ =&\{a+b\alpha+c\alpha^2+d\alpha^3+\alpha^4;a,b,c,d,e\in Z_2\} \end{align}$$

and $\alpha=x\in F$.

I do not know how I would proceed to show that the $x^2+x+1$ is irreducible over $F$. Any help would be appreciated.

Answer 1

Finite fields are remarkable in the fact that they only have one field extension (up to isomorphism) of every degree. So questions involving field extensions can often be resolved simply by working out what degree fields are involved.

$x^2 + x + 1$ is a polynomial over $\mathbb{F}_2$. Following one general procedure to solve questions such as the one you've asked, you would:

• Find the field $K$ generated by the roots of $x^2 + x + 1$.
• Find the smallest field $L$ containing both $K$ and $\mathbb{F}_{32}$
• Check the degree of the extension $L / \mathbb{F}_{32}$

None of these require doing any arithmetic with field elements. Aside from knowing $x^2 + x + 1$ is irreducible over $\mathbb{F}_2$, this exercise can be solved entirely by looking at degrees of field extensions.