I am asked to show if
- $f(x)=x^3+x+1$ over $\mathbb{F}_5$
- $g(x)=x^4+x+1$ over $\mathbb{F}_2$
are reducible or not.
I was able to show in each case they are irreducible. For $f(x)$ I assumed it is reducible and I considered $a$ to be a root. This means $a^3+a=a(a^2+1)=-1=4$. The only possibilities are $2 \times2 = 3 \times 3=4 \times 1=4$. Each would give a contradiction. Therefore, it must be irreducible.
Same reasonings apply to $g(x)$. If it is reducible then $b$ is a root and then $b^4+b=b(b^3+1)=-1=1$. The only possibility is $1 \times 1 =1$, which again gives a contradiction.
First, I'd like to know if my reasonings are fine. Secondly, I am curious to know if there are more elegant ways to solve these problems.