I can't understand my own notes in number theory….I've written: let $K$ be a number field, $\mathscr{O}_K$ its ring of integers and $U_K=\mathscr{O}_K^{\star}$ the multiplicative group of units in $\mathscr{O}_K$. Denote by $u$ the fundamental unit in $\mathscr{O}_K$, i.e. the unit greater than $1$ which generates the $\textbf{free}$ part of the units group $U_K$…..
What is $u$?! Why greater than $1$? I know Dirichlet theorem, which says that $U_K$ is the direct product of a finite cyclic group and a free abelian group of rank $r+s-1$ with $r$ the number of real embeddings and $s$ the number of pairs of non real embeddings in $\mathbb{C}$.
So, every unit $u$ in $\mathscr{O}_K$ can be expressed uniquely as $u=wu_1^{b_1}\cdots u_{r+s-1}^{b_{r+s-1}}$, with $u_j$ units in $\mathscr{O}_K$, $b_j$ integers and $w$ some root of unity.
So what is the so-called fundamental unit?
Best Answer
Let me just point you to the Wikipedia article on Fundamental units. I think it explains most of your questions so there's no point in repeating what's already in there.
Also, this mathworld article has more information and some examples.