# Find all irreducible polynomials over F of degree at most 2

## Homework Statement

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

## 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.

pasmith
Homework Helper

## Homework Statement

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

What is $\alpha^2$, and what is the characteristic of the field?

## 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.

What is $\alpha^2$, and what is the characteristic of the field?
α^2=α+1,(α+1)^2=α, the ch(F)=2

The easiest way would be to just write down all of the quadratics over this field and check whether or not each one has a root. If your question is only about monic polynomials, then there are only 16 such polynomials.

It is possible to write down a formula that counts the number of monic irreducible polynomials of a particular degree over a given finite field, and this could be used to tell you whether you had them all. However, I think that the above method would be easier in this case.