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Math Help - Easy Galois Theory Questions

  1. #1
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    Easy Galois Theory Questions

    Using permutation group notation:

    Question 1: Is e, (12)(34), (13)(24), (14)(23) a representation of C2 x C2? [Note C2 = cyclic group of order 2.]

    Question 2: Is C2 x C2 the group of symmetries of the following 4 numbers (viewed as roots of a polynomial with coefficients in Z)?
    1-sqrt(2)
    1+sqrt(2)
    1-sqrt(3)
    1+sqrt(3)

    Question 3: If the above 2 assertions are correct, then can I conclude that C2 x C2 is the Galois Group of the polynomial:
    (x^2 - 2x - 1)(x^2 - 2x - 2) = (x^4 - 4x^3 +x^2 + 6x + 2)?

    (since the first quadratic has (as roots) the first 2 numbers listed, and the 2nd
    quadratic has the latter 2)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Easy Galois Theory Questions

    Quote Originally Posted by qmech View Post
    Using permutation group notation:

    Question 1: Is e, (12)(34), (13)(24), (14)(23) a representation of C2 x C2? [Note C2 = cyclic group of order 2.]

    Question 2: Is C2 x C2 the group of symmetries of the following 4 numbers (viewed as roots of a polynomial with coefficients in Z)?
    1-sqrt(2)
    1+sqrt(2)
    1-sqrt(3)
    1+sqrt(3)

    Question 3: If the above 2 assertions are correct, then can I conclude that C2 x C2 is the Galois Group of the polynomial:
    (x^2 - 2x - 1)(x^2 - 2x - 2) = (x^4 - 4x^3 +x^2 + 6x + 2)?

    (since the first quadratic has (as roots) the first 2 numbers listed, and the 2nd
    quadratic has the latter 2)
    Why don't we work through this together? So, for Question 1 I'll assume the set of four elements of S_4 form a subgroup, call it H. We know from basic group theory that any group of order p^2 for prime p is abelian, and so H is abelian. But, by first principles (or the FTFGAG) we may conclude that H\cong \mathbb{Z}_4,\mathbb{Z}_2^2, clearly the deciding factor is whether there is an element of order 4 or not. Is there?
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  3. #3
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    Re: Easy Galois Theory Questions

    There is no element of order 4. All the permutations listed in Q1 are of order 2.
    For instance: (12)(34)(12)(34) = e.
    This multiplication (12)(34)(14)(23) = (13)(24) shows that these elements act like a group.

    I'm pretty confident about Q1 because I can multiply things out explicitly. The heart of my question is Q2 and Q3.
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