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Thread: 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 $\displaystyle Q(\zeta)=Q(\delta)$ where $\displaystyle Q(\zeta)$ is the cyclotomic field of 5th roots of unity, and $\displaystyle \delta=\frac{i}{2}\sqrt{10+2\sqrt{5}}$
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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 $\displaystyle Q(\zeta)=Q(\delta)$ where $\displaystyle Q(\zeta)$ is the cyclotomic field of 5th roots of unity, and $\displaystyle \delta=\frac{i}{2}\sqrt{10+2\sqrt{5}}$
    first see that $\displaystyle \delta$ is a root of the polynomial $\displaystyle f(x)=x^4 + 5 x^2 + 5,$ which is irreducible over $\displaystyle \mathbb{Q}$ by Eisenstein criterion. thus $\displaystyle [\mathbb{Q}(\delta): \mathbb{Q}]=4.$ since $\displaystyle [\mathbb{Q}(\zeta): \mathbb{Q}]=4,$ we'll be done if we prove that

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

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

    $\displaystyle x_k=\zeta^k - \zeta^{-k}, \ k=1,2,3,4.$ thus there exists $\displaystyle 1 \leq k \leq 4$ such that $\displaystyle \delta = x_k \in \mathbb{Q}(\zeta). \ \ \Box$
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  3. #3
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    Thank you very much for your help, NonCommAlg. So I understand that you let $\displaystyle x=\frac{i}{2}\sqrt{10+2\sqrt{5}}$. 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 $\displaystyle Q(\delta)/Q=4$

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

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

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