Let $R$ be the ring of integers in an algebraic number field. There are beautiful descriptions of $K\_0(R)$ and $K\_1(R)$. Namely, $\tilde{K}\_0(R)$ is the class group of $R$ and $K\_1(R)$ is the group of units of $R$. Question : Is there a nice description of $K\_2(R)$ (or at least some reasonable conjectures)? I couldn't find much about this in Milnor's or Rosenberg's books on algebraic K-theory, so I expect that the answer is pretty complicated. Is it maybe at least known in some special cases (say, for $R$ the integers in a quadratic extension of $\mathbb{Q}$)?



K_2 of rings of algebraic integers
It's a theorem of Garland that $K\_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence
$$0\rightarrow K\_2(R)\rightarrow K\_2(F)\stackrel{T}{\rightarrow} \oplus\_v k(v)^\*\rightarrow 0,$$
where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$ is the sum of the tame symbols, which is surjective by a theorem of Matsumoto. The injectivity on the left follows from the vanishing of $K\_2$ for finite fields. 


This isn't much of an answer, but considering $K\_2(R)$ as a subgroup of $K\_2(F)$ seems a reasonable way to start some concrete considerations. For a detailed discussion of an algorithm that proceeds essentially along these lines ('Tate's method), see the paper 


Belabas, Karim; Gangl, Herbert
Generators and relations for $K\_2( O\_F)$. 
$K$-Theory 31 (2004), no. 3, 195--231.


Added, 8 July:


I'm sure most people know this, but I forgot to mention (for newcomers) the fact that
$$K\_2(F) = F^\times\otimes\_{\mathbf Z} F^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle,$$
which I suppose motivates the original question, and makes it worthwhile to view $K\_2(R)$ as a subgroup.


