I have this doubt while studying Dirichlet unit theorem.
By dirichlet unit theorem we know that $\mathcal{U}(\Bbb{Z}[\zeta_5])=C\times F$ where $C$ is finite cyclic and $F$ is free abelian of rank $n_1+n_2-1$ where $n_1,n_2$ are real and complex embeddings of $\Bbb{Q(\zeta_5)}$ resp. Also $[\Bbb{K:Q}]=n_1+2n_2$
In the book I am reading, it is mentioned that rank of $\mathcal{U}(\Bbb{Z}[\zeta_5])=1$ but doesnt $\zeta_5$ has zero real embeddings and four complex embeddings $(\zeta_5\to {\zeta_5}^i , i\in \{1,2,3,4\})$, thus rank should be three.
If we take two complex embedding everything is correct, but there are four, no?