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Let $R$ be a domain, e.g. $\mathbb{Z}$, and let $R\to R[x]/I$ be an integral extension, e.g $\mathbb{Z}[i]$.

Is anything known about the relative algebraic $K$ groups, $K_*(R',R)$? For example, when $R\to R'$ is a nilpotent extension, the Dennis trace map from relative $K$ theory to relative negative cyclic homology is an isomorphism.

Is there a similar "simpler" spectrum that computes the relative $K$-theory of a quadratic extension?

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    $\begingroup$ I am not sure whether it is easy to compare the algebraic K-theories of real numbers and of complex numbers. $\endgroup$
    – Z. M
    Commented Nov 21 at 21:36
  • $\begingroup$ Let me mention that comparing $K_0$ should already be nontrivial: it is a direct sum of $\mathbb Z$ and the class group, and an easy control of $K_0$ should lead to, say, a simple proof of the Stark–Heegner theorem. $\endgroup$
    – Z. M
    Commented Nov 22 at 10:23

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