 Units of the fifth cyclotomic field are real up to roots of unity

Let $L=\mathbb{Q}({\zeta }_5)$ and let complex conjugation be denoted by a bar. For $u\in {\mathcal{O}}_L^{\times }$, every conjugate of $u/\bar{u}$ has modulus one, so the kernel of the logarithmic unit embedding shows that $u/\bar{u}$ is a root of unity. Reduction modulo $1-{\zeta }_5$ gives $u/\bar{u}\equiv 1$, so this root belongs to ${\mu }_5$. Squaring permutes ${\mu }_5$, hence multiplication by a suitable root of unity makes $u$ real. The real unit belongs to $\mathbb{Q}(\sqrt{5})$ and the units of the quadratic field Q square root of five finish the calculation.