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Math Help - Irreducible Factors

  1. #1
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    Irreducible Factors

    Consider the polynomial p(x)=x^6-1.

    (a) Find all monic irreducible factors of p(x) over Q.
    (b) Find all monic irreducible factors of p(x) over R.
    (c) Find all monic irreducible factors of p(x) over C.

    I'm really not sure what to do, I know that x^6 - 1 = (x^3)^2 - 1 = (x^3 - 1)(x^3 + 1) , so the the roots of p(x) are 1 & (x+1) & (x-1) are two linear factors. Anyway the first question here asks for monic (degree 1) irreducible (so that it cannot be factorized as P(x)=a(x)b(x) where a(x), b(x) \in Q) factors of p(x) over Q (rationals). Can anyone show me the method for solving one of them, so that I'll be able to try solving the rest of it on my own. Thanks.
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  2. #2
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    e^(i*pi)'s Avatar
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    x^6 - 1

    As you noted it is the difference of two squares:

    (x^3-1)(x^3+1)

    However these are the difference and sum respectively of two cubes:

    [(x-1)(x^2+x+1)][(x+1)(x^2-x+1)]

    x^2 \pm x+1 only factorises over the complex numbers.
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  3. #3
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    I'm still a little confused...

    (a) So all monic irreducible factors of p(x) over Q are:

     <br />
(x+1), (x-1), (x^2+x+1), (x^2-x+1)<br />

    (b) All monic irreducible factors of p(x) over R are:

     <br />
(x+1), (x-1), (x^2+x+1), (x^2-x+1)<br />

    (c) Now to find monic irreducible factors of p(x) over C are we use the quadratic formula for x^2 \pm x+1:

    (x^2+x+1), (x^2-x+1), \frac{-1 \pm \sqrt{-3}}{2}, \frac{1 \pm \sqrt{-3}}{2}, (x+1), (x-1)

    are all the monic irreducible factors of p(x) over complex numbers.

    Are my answers correct? Did I list all the factors correctly for each question?
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  4. #4
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    I don't know if I listed the factors correctly for each part. I'm confused...
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