Tag Archives: finite fields

The nonvanishing K_2(Z/4)

We saw previously that $K_2(F) = 0$ for a finite field $F$, where $K_2$ is the second $K$-group of $F$. It may be helpful to refer to that post for the definitions of this functor. I thought that it might be disappointing because we did all that work to compute the second $K$-group of a […]

The K-theory of finite fields: a synopsis

In my previous post, I proved that if $F$ is a finite field, then multiplicative group $F^\times$ is a cyclic group. This fact will play a small part in our description today of the $K$-theory of $F$. We will start by describing the classical $K$-theory of $F$ and then briefly talk about Quillen’s computation of […]

Number of irreducible polynomials over a finite field

Over a finite field, there are of course only finitely many irreducible monic polynomials. But how do you count them? Let $q = p^n$ be a power of a prime and let $N_q(d)$ denote the number of monic irreducible polynomials of degree $d$ over $\F_q$. The key to finding $N_q(d)$ is the following fact: the […]