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Math Help - Field extension

  1. #1
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    Field extension

    I am having trouble with this problem. It would be great if someone can help.
    I want to prove that Q(\zeta)=Q(\delta) where Q(\zeta) is the cyclotomic field of 5th roots of unity, and \delta=\frac{i}{2}\sqrt{10+2\sqrt{5}}
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  2. #2
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    Quote Originally Posted by namelessguy View Post

    I am having trouble with this problem. It would be great if someone can help.
    I want to prove that Q(\zeta)=Q(\delta) where Q(\zeta) is the cyclotomic field of 5th roots of unity, and \delta=\frac{i}{2}\sqrt{10+2\sqrt{5}}
    first see that \delta is a root of the polynomial f(x)=x^4 + 5 x^2 + 5, which is irreducible over \mathbb{Q} by Eisenstein criterion. thus [\mathbb{Q}(\delta): \mathbb{Q}]=4. since [\mathbb{Q}(\zeta): \mathbb{Q}]=4, we'll be done if we prove that

    \mathbb{Q}(\delta) \subseteq \mathbb{Q}(\zeta). now suppose \zeta=\cos a + i \sin a. then from 1=\zeta^5=(\cos a + i \sin a)^5, we'll get: 5 \cos^4 a \sin a - 10 \cos^2a \sin^3 a + \sin^5 a = \text{Im} \ (\cos a + i \sin a)^5 = 0, which after simplifying

    gives us: 16\sin^4a - 20 \sin^2 a + 5 = 0. thus: (2i \sin a)^4 + 5(2i \sin a)^2 + 5 = 0. therefore 2i \sin a=\zeta - \zeta^{-1} is a root of f(x)=0. we actually proved that the roots of f(x)=0 are exactly

    x_k=\zeta^k - \zeta^{-k}, \ k=1,2,3,4. thus there exists 1 \leq k \leq 4 such that \delta = x_k \in \mathbb{Q}(\zeta). \ \ \Box
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  3. #3
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    Thank you very much for your help, NonCommAlg. So I understand that you let x=\frac{i}{2}\sqrt{10+2\sqrt{5}}. Then squaring this equation twice and manipulate to get the polynomial. It is also the minimal polynomial correct? But I don't understand the notation you use below. Does it mean degree of Q(\delta)/Q=4

    [\mathbb{Q}(\delta): \mathbb{Q}]=4. since [\mathbb{Q}(\zeta): \mathbb{Q}]=4,
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  4. #4
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    Quote Originally Posted by namelessguy View Post

    Does it mean degree of Q(\delta)/Q=4

    [\mathbb{Q}(\delta): \mathbb{Q}]=4. since [\mathbb{Q}(\zeta): \mathbb{Q}]=4,
    yes, that's what it means. it's a standard notation for degree.
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