I need to find an irreducible polynomial in $\mathbb{F}_3$ of degree 5. To do this, I just tried some polynomials that have no roots mod 3, so if it were reducible, it would factor as an irreducible polynomial of degree 2, and another one of degree 3.
I tried the polynomial $f=x^5+x^3+x+1$ and found that $f(-1)=f(1)=f(0)=1\mod 3$ so it has no roots in $\mathbb{F}_3$. Then, I tried to divide this polynomial by the second-degree polynomials to see if it would result in a possible third-degree polynomial. The irreducible polynomials of degree 2 in $\mathbb{F}_3$ are:
- $f_1=x^2-x-1$
- $f_2=x^2+x-1$
- $f_3=x^2+1$
Dividing by $f_1$ gives $x^3+x^2+1$ but then leaves a rest term of $2x+2$ in long division: $(x^5+x^3+x+1)-(x^2-x-1)(x^3+x^2+1)=2x+2$
Similarly, dividing by $f_2$ gives $f-f_2(x^3-x^2-1)=2x+2$, and dividing by $f_3$ gives $f-f_3*x^3=x-2$.
From this, I concluded that $f$ is not divisible by any polynomial of degree 2 in $\mathbb{F}_3$, and therefore cannot split, but I was not sure if this is a solid argument. Besides, this seems like a bit of a random way to find such a polynomial, so any other (simpler) approach to find an (any) irreducible polynomial of degree 5 in $\mathbb{F}_3$ would be greatly appreciated.