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Irreducible polynomials

  1. Sep 19, 2012 #1
    If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?
     
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  3. Sep 19, 2012 #2

    Dick

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    Re: Irreduceable polynomials

    I wouldn't drag imaginary numbers into this. If x^2+1 is reducible over the integers then it would split into two linear factors with integer coefficients. Can you argue why that can't happen?
     
  4. Sep 20, 2012 #3
    Re: Irreduceable polynomials

    is it because x^2+1 has no real solutions?
     
  5. Sep 20, 2012 #4

    HallsofIvy

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    Strictly speaking, because the problem said "irreduceable over the integers", it is because there are no integers satisfying [itex]x^2+1= 0[/itex]. Of course, since the integers are a subset of the real numbers yours is a sufficient answer. But your teacher might call your attention to the difference by (on a test, perhaps) asking you to show that [itex]x^2- 2[/itex] is irreduceable over the integers.
     
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