Find all irreducible polynomials over F of degree at most 2

  • #1
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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.
 

Answers and Replies

  • #2
pasmith
Homework Helper
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Homework Statement


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

What is [itex]\alpha^2[/itex], and what is the characteristic of the field?

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.
 
  • #3
85
0
What is [itex]\alpha^2[/itex], and what is the characteristic of the field?
α^2=α+1,(α+1)^2=α, the ch(F)=2
 
  • #4
52
7
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.
 

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