Hi all, and apologies for the delay in posts. I was participating in the Intel Science Talent Search, which was in Washington, DC. In some of my questions, I was asked to compute the algebraic K-theory of finite fields, and I wanted to give an outline of this awesome computation.
Let denote the finite field with elements. We’ll outline the computation of , but in order to do that, we need some homotopy theory. The Adams operations on are given by the adams operations , in the sense that the Adams operations on are given by . Quillen proved the following theorem:
Theorem (Quillen): The space is the homotopy fiber of .
How does this help us? Well, is just multiplication by on , because is simply . Consequently, we get the following computation of the K-groups of :
The K-groups of : for even, . For odd, . (Quillen also showed that if then the induced map is injective.)
There are also interesting computations for mod coefficients. In fact, Browder showed that is isomorphic as a graded ring to , where and are generators for and respectively ( is called the Bott element). If and , then is the union of the , and so by the second part of the computations of the ordinary K-groups (which also holds mod ), we see that is the direct limit of the , and it turns out that is isomorphic to .
I apologize for the short post after the lengthy silence, but I will try to post more regularly soon, now that “senior” year is slowing down.