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Math Help - minimum polynomial

  1. #1
    Junior Member ibnashraf's Avatar
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    minimum polynomial

    Can someone help me with the following question please ...

    (i) Let \alpha=\sqrt(\sqrt5+\sqrt15) . Find the minimum polynomial of \alpha over Q.
    (ii) Give a basis for Q( \alpha) over Q.
    (iii) What is [Q( \alpha):Q] ?

    So far for the minimum polynomial I got (\alpha^8)-40(\alpha^4)+100=0. Is this correct?

    How do I find the required basis?

    Thanks.
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  2. #2
    MHF Contributor

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    Re: minimum polynomial

    i suggest taking the scenic route:

    the polynomial you gave is ONE annihilating polynomial for α, but how do you know it is minimal (i.e. irreducible)?

    step 1) show α is not in Q(β), where β = √5 + √15. this means that [Q(α):Q(β)] ≥ 2.

    step 2) show Q(β) = Q(√3,√5) (hint: Q(β) is obviously contained in Q(√3,√5), so show that both √3,√5 are contained in Q(β)). this shows that [Q(β):Q] = 4.

    step 3) deduce from steps 1&2 that [Q(α):Q] is at least 8, and therefore *is* 8. therefore x8 - 40x4 + 100 is irreducible over Q, and is therefore the minimal polynomial for α.

    note that in the course of answering (i), (iii) is taken care of.

    for part (ii), show that the first seven powers of α, and 1 are linearly independent. why is this enough? (hint: linear dependence means α satisfies a polynomial of degree....?)
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