Prove that $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$ is the ring of integers of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$

Question

I'm a bit at a loss about what I can say in this situation. Do I have to show that $\zeta_{p} + \zeta_{p}^{-1}$ form an integral basis ? If I do, I have no idea how to do it.

If not, can I use the fact that $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1}) \subseteq \mathbb{R}$ at my advantage ?

Thanks in advance for any hint or answer,

Jérôme

Answer 1

Let $\tau_p=\zeta_{p} + \zeta_{p}^{-1}$ then since $\bar{\zeta_p}=\zeta_{p}^{-1}$ we have that $\tau_p \in \mathbb{R}$ so that $\mathbb{Q}(\tau_p) \subset \mathbb{R}$. We know that $\{\zeta,\zeta^2,\ldots,\zeta^{p-1}\}$ form an integral basis for the cyclotomic integers. Let $a=\sum_{i=1}^{p-1}b_i\zeta_p^i$ be a real cyclotomic integer, then $a=\bar{a}$. Since $\bar{\zeta_p^i}=\zeta_p^{p-i}$ we must have $b_i=b_{p-1}$ but then $a=\sum_{i=1}^{(p-1)/2}\tau_p^i$ which shows that the $\{\tau,\tau^2,\ldots,\tau^{(p-1)/2}\}$ form an integer basis of the real quaternion integers.

Answer 2

You show that every element of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$ which is a root of a monic integer polynomial is actually in $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$.