nt.number theory - Two questions about units in Number Fields

Dirichlet's unit theorem is a relatively straightforward extension of the well known proof that the Pell equation has a nontrivial solution by a clever use of Dirichlet's box principle. The "machinery" is only needed to control the combinatorial explosion, or,
as far as geometry of numbers is concerned, as a natural generalization of the box principle.

As for your second question I first thought that the answer is no. Take for example a biquadratic number field $K = {mathbb Q}(sqrt{m},sqrt{n},)$, with $m,n > 0$ chosen in such a way that the unit index (units of K : units from the subfields) is 1. This implies that the only units in K are multiples of those coming from the three subfields. But you still get generators of the field by taking the product of two units coming from different subfields, so your question is still open.