I am reading the part of Washington's book related to Thaine's theorem. Let $m$ be a positive integer and let $F$ be the maximal real subfield of $\mathbb Q(\zeta_m)$, i.e. $F=\mathbb Q(\zeta_m+\zeta_m^{-1})$. Moreover let $l$ be a prime congruent to $1$ modulo $m$.
Then Washington defines an element $\epsilon=\prod_\gamma\prod_a(\zeta_m^{a\gamma}-\zeta_l)^{b_a}$ where $\gamma$ runs through $Gal(\mathbb Q(\zeta_{ml}|F(\zeta_l))$. He says that since $(m,l)=1$ each factor is a unit in $\mathbb Z[\zeta_{ml}]$ and so $\epsilon$ is a unit of $F(\zeta_l)$. Why is this true?
I agree that each factor is a unit in $\mathbb Z[\zeta_{ml}]$ but I cannot see how this implies that $\epsilon$ is a unit in $F(\zeta_l)$. Is there a more general property ? Please help, honestly I do not see even the logic in this claim. Thank you very much.
Since each factor is a unit in $\mathbb{Z}[\zeta_{ml}]$, so is $\epsilon$. So let $\epsilon' \in \mathbb{Z}[\zeta_{ml}]$ be such that $\epsilon \epsilon' = 1$. Let $G = Gal(\mathbb{Q}(\zeta_{ml})/F(\zeta_l))$. Let $\epsilon_1 = \prod_a (\zeta_m^{a}-\zeta_l)^{b_a}$. The conjugates of $\epsilon_1$ under the action of $G$ are precisely the $\prod_a (\zeta_m^{a\gamma}-\zeta_l)^{b_a}$'s where $\gamma$ ranges over elements in $G$, so $\epsilon$ is invariant under $G$-action. Therefore $\epsilon'$ is also invariant under $G$-action, so $\epsilon' \in F(\zeta_l) \cap \mathbb{Z}[\zeta_{ml}]$, and this intersection is the ring of integers of $F(\zeta_l)$.