From the works of Borel and Quillen there is a connection between the $K$-theory of the ring of integers $\mathfrak{o}_K$ in a number field $K$ and the arithmetic of the number field. In fact, it is well known that

$K_0(\mathfrak{o}_K)=\mathbb{Z}\oplus \mathrm{Cl}(\mathfrak{o}_K)$, where $\mathrm{Cl}(\mathfrak{o}_K)$ denotes the class group of $K$;

$K_1(\mathfrak{o}_K)=\mathfrak{o}_K^\times$ (group of units in $\mathfrak{o}_K$);

For $n\geq 2$ all the $K$-groups are finitely generated:

- $K_{2n}(\mathfrak{o}_K)$ is a finite group for $n\geq 1$;
- $K_{2n-1}(\mathfrak{o}_K)= \mathbb{Z}^{d_n}\oplus (\text{finite group})$ where $d_n=r_1+r_2$ or $r_2$ (the number of real/complex embeddings), depending on if $n\equiv 1 \mod 4$ or $n\equiv 3\mod 4$.
- Also, and here comes the pretty remarkable meat of the story: for $n=2m-1$, the order of vanishing of the Dedekind $\zeta$-function, $\zeta_K(s)$, at $s=1-m$ is equal to the rank of $K_{2m-1}$.

(All this can be found in Weibel's survey in the $K$-theory handbook, or Kolster's survey from ICTP; google and ye shall find.)

Now, my questions are:

1) Since the definitions of the $K$-groups are pretty topological in nature, in addition quite complicated (and I am very far from being a topologist), is there any 'reason' why this connection between $K$-theory and the arithmetic of number fields exists at all. It seems like a complete miracle. In other words, why *should* this connection exist?

2) What is the arithmetic significance of the torsion parts of the above $K$-groups? There is some information on this in Weibel's survey mentioned above, but I don't understand what he is talking about.

Admittedly, question 1) is vague, but I hope it's clear what I'm after.

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