Is $f(x)=x^4+x+2$ irreducible over $\mathbb{Z}_3$?

Question

I am looking for irreducible polynomials to construct finite fields. In this case I need a degree $4$ irreducible polynomial over $\mathbb{Z}_3$. Is $f(x)=x^4+x+2$ irreducible over $\mathbb{Z}_3$? I don't have much practice proving the irreducibility of a polynomial. So I am not sure what I have done. It's okay?

First, I have that $f(x)$ has no roots in $\mathbb{Z}_3$. If $f(x)$ was reducible, noting that there are no $x^3$ term in $f(x)$. So the factorization could be $(x^2+2x+c)(x^2+x+d)$, where $c,d \in \mathbb{Z}_3$ ($2 \equiv -1 \pmod 3$). So, $$x^4+x+2=x^4+(c+d+2)x^2+(c+2d)x+cd$$ Then $c+d+2=0$, $c+2d=1$ and $cd=2$. But of the first two equalities I have to $d=0$, wich is impossible since $cd\neq 0$. Thus $f(x)$ is irreducible over $\mathbb{Z}_3$.

Answer 1

What you did is fine. Here is a conceptual overkill. First as $f(x)$ has no root in $\mathbb F_3$, if it's reducible, all its roots must be in $\mathbb F_9$, and the Galois group $G(\mathbb F_9/\mathbb F_3) = \{1, \rho\}$ where $\rho(x) = x^3$ is the Frobenius element. In particular, if $x\in \mathbb F_9\setminus \mathbb F_3$, its unique Galois conjugate is given by $\rho(x)=x^3$, therefore $x^4=x\cdot\rho(x)\in\mathbb F_3$. From the equation $x^4+x+2=0$, we get $x = -2 - x^4\in\mathbb F_3$, but we have already said $x\not\in\mathbb F_3$.

If you don't know Galois theory yet, note that $\mathbb F_9^{\times}$ is a (cyclic) group of order $8$, hence $x^8=1$ and $x^4=\pm 1\in\mathbb F_3$.

Answer 2

Assume that we have a factorization

$$x^4 + x + e= (x^2 + a x + b)(x^2 + c x + d)$$

Looking at the coefficient of $x^3$ we get $c=-a$. Now the coefficient of $x^2$ on RHS is $-a^2 + b + d$ so $d = a^2- b$. Now we get $$(x^2 + a x + b)(x^2 - a x + (a^2 - b))= x^4 + (a^3 - 2 a b)x + a^2 b - b^2 $$

Now we get the system $$a^3 - 2 a b = 1 \\ a^2 b - b^2 = e $$

Express $b$ from the last equation (assume $2$ invertible) and substitute in the second one. We get $$a^6 - 4 e a^2 -1 = 0$$

Now in our case $e=2$, so the equation is $a^6 - 8 a^2 - 1 = 0$. Moreover, over $\mathbb{F}_3$ we have $x^3 = x$ for all $x$, so $a^6 = a^2$. We get the equation $a^2 = -1$, not possible.