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Irreducible Polynomial over Z

  1. Dec 9, 2008 #1
    I am trying to figure out if the polynomial [(x^4)+1] is reducible over Z5 and also over Z.

    For Z5, i tried:
    f(0) = 1
    F(1) = 2 = f(2) = f(3) = f(4)
    Since neither are zero, i tried
    f(x) = (ax^2 + bx + c)(dx^2 + ex + f)

    I compared the coefficients but am unable to solve it.

    For Z, i have no idea how to do it.

    Please help! Thanks.
     
  2. jcsd
  3. Dec 9, 2008 #2
    Say a = d = 1, so suppose x4 + 1 = (x2 + bx + c)(x2 + ex + f). Comparing coefficients gives:

    (x3) 0 = b + e
    (x2) 0 = c + f + be
    (x1) 0 = bf + ce
    (x0) 1 = cf.

    Fiddling around should give either a contradiction somewhere, or some factorization. Now 0 = b + e, so b = -e. Thus the x2 part gives 0 = c + f - b2, so b2 = c + f.

    Working in Z, cf = 1, so c = f = ±1 and c + f = ±2. But this is a contradiction, since b2 = c + f and c + f is not a perfect square. Many irreducibility proofs follow a similar pattern.

    In Z5, since cf = 1, c + f is 0, 2, or 3 (by looking at each possible pair of c, f). But c + f = b2, so c + f = 0 (since 2 and 3 are not perfect squares), so since cf = 1 as well, you have c, f = 2, 3 in some order. Also, b = -e = 0 since c + f = 0 = b2. You can check that x4 = (x2 + 2)(x2 + 3).
     
    Last edited: Dec 9, 2008
  4. Dec 10, 2008 #3

    Office_Shredder

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    I'm not sure what theorems you have for Z, but you can use Eisenstein's Criterion for this (which works more generally also) by noting:
    (1)f(x) is irreducible if and only if f(t+a) is irreducible for any constant a (so you're changing variables to t by the transformation x=t+a)..... note that f(x) = g(x)h(x) if and only if f(t+a) = g(t+a)h(t+a)

    (2) Substitute t+1 for x in the original polynomial

    (3) note that 4Ck (4 choose k) is divisible by 2 for all k not 0 or 4. Hence the coefficients in the new polynomial satisfy Eisenstein's Criterion
     
  5. Dec 11, 2008 #4
    Thanks for the help. greatly appreciate it.
     
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