How Do You Determine Irreducible Polynomials Over Finite Fields?

[mathusers]

(1):
Find all irreducible polynomials of the form $x^2 + ax +b$, where a,b belong to the field $\mathbb{F}_3$ with 3 elements.
Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [$\mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.

any suggestions please?

 

[jacobrhcp]

no, but I am currently doing problems that look a lot like this. I would really enjoy seeing this problem solved. =).

 

[Hurkyl]

(1):
Find all irreducible polynomials of the form $x^2 + ax +b$, where a,b belong to the field $\mathbb{F}_3$ with 3 elements.
Show explicitly that $\mathbb{F}_3(x)/(x^2 + x + 2)$ is a field by computing its multiplicative monoid.
Identify [$\mathbb{F}_3(x)/(x^2 + x + 2)$]* as an abstract group.

any suggestions please?

It's a very small problem. Have you tried brute force?