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Irreducible Polynomial of Degree 3

  1. Aug 20, 2014 #1
    1. The problem statement, all variables and given/known data
    If p(x) ∈F[x] is of degree 3, and p(x)=a0+a1∗x+a2∗x2+a3∗x3, show that p(x) is irreducible over F if there is no element r∈F such that a0+a1∗r+a2∗r2+a3∗r3 =0.


    2. Relevant equations



    3. The attempt at a solution
    Is this approach correct?
    If p(x) is reducible, then there exists ax + b such that a, b ε F and a≠0. And p(x) = (ax + b)(cx^2 + dx + e). Then an r exists such that p(r) = 0.

    Thank you.
     
  2. jcsd
  3. Aug 22, 2014 #2

    haruspex

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    Yes, though you could go into a bit more of an explanation as to why there would have to be a first degree factor.
     
  4. Aug 22, 2014 #3

    Mark44

    Staff: Mentor

    With over 300 posts in this forum, you should have learned enough of the ropes here to write exponents clearly.

    At the very least, use ^ to indicate exponents, as you do below. Even nicer would be to use the exponent button from the advanced menu - click Go Advanced to show this menu, and then click the X2 to make exponents.

    I think what you meant above was p(x) = a0 + a1x + a2x2 + a3x3 = 0, and similarly for your other equation.
     
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