How to Determine Irreducible Polynomials in (Z/2Z)[x]?

[silvermane]

Homework Statement

Find all monic polynomials of degrees 2 and 3 in (Z/2Z)[x]. Determine which ones are irreducible, and write the others as products of irreducible factors.

The Attempt at a Solution

I know that factors of degree 1 correspond to roots in Z/2Z and that monic polynomials are polynomials where the top term coefficient is equal to 1. I think I'm not understanding what I'm supposed to do, or how to write them via modulus.

Any hints or tips are greatly appreciated! :)
Please no answers!

 

[micromass]

Here are the polynomials of dregree 2:

x²
x²+1
x²+x
x²+x+1

Can you now find all the polynomials of degree 3? There are 8 of them.

 

[silvermane]

So It's pretty much just combinations of functions with an X^2 for degree 2, and other terms such as x and 1 for the lesser degrees.

Furthermore, for x^3, we would have this:

x^3
X^3 +x^2
X^3 +x^2+x
X^3 +x^2+x+1
X^3 +x+1
X^3 +x^2+1
X^3 +1
X^3 +x

Thanks for your help!
I think I can handle it from here :)

 

[micromass]

Yes!

Good luck!