TITLE: Two questions about units in Number Fields
QUESTION [14 upvotes]: If $K/\mathbb{Q}$ is a number field which is not $\mathbb{Q}$ or a quadratic imaginary field then, by the Dirichlet unit theorem it has a unit of infinite order.  Is there a simple proof of this fact which doesn't haul in all the machinery of the unit theorem?  In a related question is it true that such a $K$ is always generated over $\mathbb{Q}$ by one unit (which might not be of infinite order -- cf. the case of a full cyclotomic field), and if it is, is there a simple proof of this fact?

REPLY [11 votes]: The answer to question 2 as stated is no. Let $K = \mathbb{Q}(\sqrt{p},\sqrt{-b})$, with $p$ 1 mod 4. By Frohlich and Taylor, p. 196, the fundamental unit group is generated by a unit from the real quadratic subfield. It then suffices to show that we can pick b so that K contains no roots of unity other than $\pm 1$. But the only possibility is that K contains a 3rd, 4th, or 5th root of unity, and so we just pick $b$ to avoid ramification at 2, 3, or 5.