# Irreducible Polynomials Over A Field

• Jan 20th 2011, 07:02 PM
Shapeshift
Irreducible Polynomials Over A Field
Are the polynomials in f(x) = x^3 + 2x^2 + 1 and g(x) = x^2 + x + 1 irreducible in F5[x]? Why or why not?

I'm having trouble checking to see if they are irreducible or not. I know it has something to do with the roots but I'm not too sure. Any help would be appreciated.
• Jan 20th 2011, 07:22 PM
tonio
Quote:

Originally Posted by Shapeshift
Are the polynomials in f(x) = x^3 + 2x^2 + 1 and g(x) = x^2 + x + 1 irreducible in F5[x]? Why or why not?

I'm having trouble checking to see if they are irreducible or not. I know it has something to do with the roots but I'm not too sure. Any help would be appreciated.

Hint: Over any field, a polynomial of degree less than or equal 3 (three) is irreducible iff it has no root in the field.

Tonio
• Jan 20th 2011, 07:27 PM
Shapeshift
So am I right in thinking that both f(x) and g(x) are irreducible since none of their roots = 0 (mod 5)?
• Jan 20th 2011, 07:30 PM
tonio
Quote:

Originally Posted by Shapeshift
So am I right in thinking that both f(x) and g(x) are irreducible since none of their roots = 0 (mod 5)?

I supose you meant "...since none of the elements of the field $\mathbb{F}_5$ is a root of any of them", and then yes: you'd be right.

Tonio
• Jan 20th 2011, 07:32 PM
Shapeshift
Haha sorry, I'm new to this concept so I'm still having a little trouble truly understanding it. Thank you!