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Math Help - Irreducible polys

  1. #1
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    Irreducible polys

    THis is really long but wanted to check my work

    FInd all eighth-roots of 1. Use this to express f (x)=x^8-1 as a product of irreducible polynomials
    a) in C [x]
    b) in R [x]
    c) in Q [x]
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    THis is really long but wanted to check my work

    FInd all eighth-roots of 1. Use this to express f (x)=x^8-1 as a product of irreducible polynomials
    a) in C [x]
    b) in R [x]
    c) in Q [x]
    As 1=exp(2 pi n i) n=0, +/-1, +/-2, .. , the 8th roots of unity are:

    r(n) = exp(2 pi (n/8) i) n=0, +/-1, +/-2, .. ,

    which give 8 distinct values for n=0, 1, 2, .., 7, with r(0)=1, and r(4)=-1
    being the real roots, all the others are complex.

    RonL
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  3. #3
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    Hello, chadlyter!

    Find all eighth-roots of 1.
    Use this to express f (x) = x^8 -1 as a product of irreducible polynomials

    a) in C[x] . . b) in R[x] . . c) in Q[x]
    The problems are easier in reverse order.


    (c) Factor the polynomial.

    x^8 - 1 .= .(x^4 - 1)(x^4 + 1) .= .(x - 1)(x + 1)(x^4 + 1)

    . . = .(x - 1)(x + 1)(x + 1)(x^4 + 1)


    (b) The quartic factor can be factored:
    . . . . . . . . . . . . . . . . . . . . . . - . . - . ._ . . . . . . . . . _
    x^8 - 1 .= .(x - 1)(x + 1)(x + 1)(x - √2x + 1)(x + √2x + 1)

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._
    (a) The eight roots are: . 1, i, (1 i)/√2

    To save me typing all those ugly radicals,
    . . let: .x
    1 .= .(1 + i)/√2, .x2 .= .(1 - i)/√2, .x3 .= .(-1 + i)/√2, .x4 .= .(-1 - i)/√2

    x^8 - 1 .= .(x - 1)(x + 1)(x - i)(x + i)(x - x
    1)(x - x2)(x - x3)(x - x4)

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