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## Homework Statement

Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2.

## Homework Equations

## The Attempt at a Solution

To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F,

So far, I got: x^2+x+α,x^2+x+α+1,x^2+αx+1,x^2+αx+α,x^2+(α+1)x+1,x^2+(α+1)x+α+1

these polynomials don't have any roots in F (if my calculation right), but I am not sure that I have all irreducible polynomial or not. Can someone check for me or provide an easy way to me so that I can check by myself? Thanks.