Exercise 4, § 2

Let $K$ be the smallest Pythagorean subfield of $\mathbf{R}$ and let $K' = K(i)$. Let $G$ be the multiplicative group of elements of $K'$ of absolute value 1 (a subgroup of $\mathbf{U}$). Show that $G$ is not isomorphic to the additive group of numbers of $K$ mod 1. (Observe that in the latter group there exist elements of any prime order $p$; on the other hand, if $p$ is a prime such that $p - 1$ is not a power of 2, show that $G$ contains no $p$th root of unity other than 1, by noting that the degree over $\mathbf{Q}$ of every element of $K'$ is a power of 2.)