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Math Help - UFD or Not?

  1. #1
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    UFD or Not?

    In Z_5[x] we have x^2 +3x +1 = (2x+3)(3x+2) but we also have x^2 +3x +1 = (4x+1)(4x+1) does this mean that Z_5[x] is not a UFD?
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  2. #2
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    Quote Originally Posted by Coda202 View Post

    In Z_5[x] we have x^2 +3x +1 = (2x+3)(3x+2) but we also have x^2 +3x +1 = (4x+1)(4x+1) does this mean that Z_5[x] is not a UFD?
    no! for any field F, the ring F[x] is always a UFD (even more, it's a PID). in your questions, modulo 5 we have:

    2x+3=2x - 2=2(x-1) and 4x + 1 = -x +1. also we have 3x+2=-2x + 2 = 2(-x+1).

    as you see the irreducible factors 2x+3 and 3x+2 are equal to 4x+1 multiplied by a unit. this does not contradict the definition of a UFD.
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