# Math Help - irreducible polynomials

1. ## irreducible polynomials

Show that the polynomials $g = X^3 + X + 1$ and $h = X^3 + X^2 + 1$ 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 $\leq 3$ and have no roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

Thanks in advance!

2. Hello,

Originally Posted by hunkydory19
Show that the polynomials $g = X^3 + X + 1$ and $h = X^3 + X^2 + 1$ 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 $\leq 3$ and have not roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

Thanks in advance!
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

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