# Irreducible Polynomials:

• Mar 3rd 2008, 09:37 AM
joanne_q
Irreducible Polynomials:
(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.

• Mar 3rd 2008, 10:07 AM
ThePerfectHacker
Quote:

Originally Posted by joanne_q
(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.

Just list all of them $x^2 + x+1,x^2+x+2,...$ there are only $9$ or them. Now, just check which ones have zeros (there are only three zeros to check). And this tells you which are irreducible and which are not.

Quote:

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 element in $\mathbb{F}_3[x]/(x^2+x+2)$ has form $a+bx+(x^2+x+2)$. Now show that these elements form a field.