irreducible polynomials

**hunkydory19** · May 5th 2008, 02:34 AM

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!

**Moo** · May 5th 2008, 02:38 AM

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

**Moo** · May 5th 2008, 02:44 AM

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.

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