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Find all irreducible polynomials over F of degree at most 2

  1. Feb 17, 2016 #1
    1. The problem statement, all variables and given/known data
    Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2.

    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Feb 18, 2016 #2

    pasmith

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    Homework Helper

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

     
  4. Feb 18, 2016 #3
    α^2=α+1,(α+1)^2=α, the ch(F)=2
     
  5. Feb 24, 2016 #4
    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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