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 $A_0$.
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Best Answer
As Dylan points out, parts (1) and (2) are clear. Moreover, $\mathbb{Z}[\zeta + \zeta^{-1}]$ contains $\zeta^j + \zeta^{-j}$ for all $j \ge 1$ (by induction using the binomial theorem); these include all the conjugates of $\zeta + \zeta^{-1}$, so (4) implies (3). Thus it suffices to prove (4), which follows from the corresponding fact for the full cyclotomic field $\mathbb{Q}(\mu_\ell)$ (which is well known), as follows:
Let $u \in A_0$. Because $u$ is an algebraic integer in $\mathbb{Q}(\zeta)$, we can write $u = \sum_{i = 0}^{\ell - 1} u_i \zeta^i$ for some $u_i \in \mathbb{Z}$. But since $u = \overline{u}$, we have $u = \sum u_i \zeta^{-i} = \sum u_i \zeta^{\ell - i}$. Hence $u_i = u_{\ell - i}$. Thus we have $u = \sum_{i = 0}^{(\ell - 1)/2} u_i (\zeta_i + \zeta^{-i})$, and (4) is proved.