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Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$.

My question: Is the following proposition true? If yes, how would you prove this?

Proposition

(1) $K_0 = \mathbb{Q}(\zeta + \zeta^{-1})$ is the maximal real subfield of $K$.

(2) $[K_0 : \mathbb{Q}] = (l - 1)/2$

(3) The ring of algebraic integers $A_0$ in $K_0$ is $\mathbb{Z}[\zeta + \zeta^{-1}]$.

(4) $\zeta + \zeta^{-1}$ and its conjugates constitute an integral basis of $A_0$.

Related questions

This and this.

  • 2
    You can get some of this cheaply, I think. (1) and (2) should follow from the fact that $\zeta + \zeta^{-1}$ is fixed by conjugation and the fact that $\mathbb Q(\zeta)$ will split the quadratic $(X - \zeta)(X - \zeta^{-1}) = X^2 - (\zeta + \zeta^{-1})X + 1$.2012-07-25
  • 0
    Ah, right, sorry for botching the title. Just wanted to be a bit more descriptive.2012-07-25
  • 0
    No problem, Dylan. Thanks.2012-07-25

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