Let $K=GF(2)$ and $p(x)= x^3 + x+1.$ Show that $p$ is irreducible in $K[x]$

Question

Let $K=GF(2)$ and $p(x)= x^3 + x+1$ Show that $p$ is irreducible in $K[x]$

First of all am I right in interpreting:

$$GF(2) = \mathbb Z / 2 \mathbb Z= \{ 0,1\}$$

So basically, $p(x)$ is a polynomial with coefficients in $K$, right?

Does that mean if $$p(x) = a_0 + a_1 x + a_2 x^2 + .... + a_n x^n$$ and $ p(x) \in K[x]$ means $ K = \{a_0, a_1, a_2,...,a_n\}$, that the coefficients of $p$ are going to come from this tiny set? $\{0,1\}$

Or am I supposed to find the possibilities of $x$ in this set, $\{0,1\}$ because in that case $p(x)$ is irreducible in $K$

Answer 1

If $P$ was reducible, it would be the product of a polynomial of degree $1$ multiplied by a polynomial of degree $2$. And a polynomial of degree $1$ has a root in the base field.

However your polynomial is never vanishing on $K$.

Now answering the question in the comment of Siyanda.

$GF(8)$ is the only finite field with $8$ elements and its dimension over $GF(2)$ is equal to $3$. So $GF(8) \simeq K[X]/(P)$ which means that $P$ has a root in $GF(8)$.

Answer 2

Oven a finite field $\mathbb{K}$, a third-degree polynomial is irreducible iff it has no roots in $\mathbb{K}$.

Given that, it is trivial that the only irreducible third-degree polynomials over $\mathbb{F}_2$ are $x^3+x^2+1$ and $x^3+x+1$.