# Irreducible Polynomials:

• Mar 3rd 2008, 09:37 AM
joanne_q
Irreducible Polynomials:
(1):
Find all irreducible polynomials of the form \$\displaystyle x^2 + ax +b \$, where a,b belong to the field \$\displaystyle \mathbb{F}_3\$ with 3 elements.
Show explicitly that \$\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)\$ is a field by computing its multiplicative monoid.
Identify [\$\displaystyle \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 \$\displaystyle x^2 + ax +b \$, where a,b belong to the field \$\displaystyle \mathbb{F}_3\$ with 3 elements.

Just list all of them \$\displaystyle x^2 + x+1,x^2+x+2,...\$ there are only \$\displaystyle 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 \$\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)\$ is a field by computing its multiplicative monoid.
Identify [\$\displaystyle \mathbb{F}_3(x)/(x^2 + x + 2)\$]* as an abstract group.
Any element in \$\displaystyle \mathbb{F}_3[x]/(x^2+x+2)\$ has form \$\displaystyle a+bx+(x^2+x+2)\$. Now show that these elements form a field.