Our freedom does not lie outside us, but within us. One can be bound outside, and yet one will still feel free since one has burst inner bonds. One can certainly gain outer freedom through powerful actions, but one creates inner freedom only through the symbol.
The symbol is the word that goes out of the mouth, that one does not simply speak, but that rises out of the depths of the self as a word of power and great need and places itself unexpectedly on the tongue. It is an astonishing and perhaps seemingly irrational word, but one recognizes it as a symbol since it is alien to the conscious mind. If one accepts the symbol, it is as if a door opens leading into a new room whose existence one previously did not know. But if one does not accept the symbol, it is as if one carelessly went past this door; and since this was the only door leading to the inner chambers, one must pass outside into the streets again, exposed to everything external. But the soul suffers great need, since outer freedom is of no use to it. Salvation is a long road that leads through many gates. These gates are symbols. Each new gate is at first invisible; indeed, it seems at first that / it must be created, for it exists only if one has dug up the spring’s root, the symbol.P.311, The red book
[Jung] Jung, C. G., Shamdasani, S. E., Kyburz, M. T., & Peck, J. T. (2009). The red book: Liber novus. WW Norton & Co.
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.
This is an outline of Quillen’s proof for the calculation of K-theory of finite fields, originally done by Quillen in , see also  for a slightly different presentations with more background materials included.
Here are the full post:
 Quillen, Daniel. “On the cohomology and K-theory of the general linear groups over a finite field.” Annals of Mathematics 96.3 (1972): 552-586.
 Mitchell, S. Notes on K theory of nite elds. Available online: