Starting with some motivations and brief expositions on algebraic K theory, I’ll introduce some early important computations of algebraic K-theory, including computations of K theory of finite fields and of rings of integers for which I will briefly outline the proofs. Then we’ll move on to K-theory with finite coefficients of separably closed fields. With the motivation of recovering some information of K-theory of an arbitrary field from its separable closure, we introduce a few versions of the Lichtenbaum-Quillen conjectures as descent spectral sequences of \’etale Cohomology groups. If time permits, I’ll mention relation to motivic Cohomology that a key tool is some “motivic-to-K-theory” spectral sequence.
These are notes based on my talk on Oct 01, 2021 in UIUC Graduate Homotopy Seminar. The main references are  and . The outline of the proof of Quillen’s K-theory of finite fields has been moved to Appendix A.
Here are the notes, updated on 10/30/2021. I thank Prof Grayson for comments and pointing out some typos.
Updates (10/30/2021): Fixing a few typos, adding reference to the claim about the fixed point spectrum for a Galois extension .
 Mitchell, Stephen A. “On the Lichtenbaum-Quillen conjectures from a stable homotopy-theoretic viewpoint.” Algebraic topology and its applications. Springer, New York, NY, 1994. 163-240.
 Weibel, Charles A. The K-book: An introduction to algebraic K-theory. Vol. 145. Providence, RI: American Mathematical Society, 2013.