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I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is: What is the list of rings such that all their algebraic $K$-theory groups are known? I ...

With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$? Here, by "trivial examples" ...