# irreducible polynomials

• May 5th 2008, 02:34 AM
hunkydory19
irreducible polynomials
Show that the polynomials $\displaystyle g = X^3 + X + 1$ and $\displaystyle h = X^3 + X^2 + 1$ are the only polynomials of degree 3 which are irreducible over F2.

I know that they are both irreducible over F2 since they both have degree $\displaystyle \leq 3$ and have no roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

• May 5th 2008, 02:38 AM
Moo
Hello,

Quote:

Originally Posted by hunkydory19
Show that the polynomials $\displaystyle g = X^3 + X + 1$ and $\displaystyle h = X^3 + X^2 + 1$ are the only polynomials of degree 3 which are irreducible over F2.

I know that they are both irreducible over F2 since they both have degree $\displaystyle \leq 3$ and have not roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

Because the other polynomials of degree 3 in F2 are x^3, x^3+1, x^3+x^2 and x^3+x and that they all have a root in F2 :D
• May 5th 2008, 02:44 AM
Moo
Quote:

since they both have degree $\displaystyle \leq 3$
Hey watch out. Degree 3 doesn't mean degree $\displaystyle \leq 3$. Polynomials have to contain the term $\displaystyle X^3$ here.