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Math Help - Fields

  1. #1
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    Fields

    let a= cuberoot(2+sqrt(6)), what does the field Q(a) look like?
    If B=a^2 +a +2, then express B^-1 in the form alpha_5(a^5)+ alpha_4(a^4)+...+alpha_0 where the alpha_i's exist in Q.
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    let a= cuberoot(2+sqrt(6)), what does the field Q(a) look like?
    If B=a^2 +a +2, then express B^-1 in the form alpha_5(a^5)+ alpha_4(a^4)+...+alpha_0 where the alpha_i's exist in Q.
    If a=\sqrt[3]{2+\sqrt{6}} then a^3 - 2 = \sqrt{6} so a^6 - 2a^3 - 2 = 0. Thus, a is root of polynomial x^6 - 2x^3 - 2. This polynomial is irreducible so \mathbb{Q}(a) are all elements of form a_0 + a_1 a+ ... + a_5a^5 for a_i \in \mathbb{Q}.
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