Number theory: K_0 and K_1 of a ring of integers
Let O_K be the ring of integers of a number field K. Two classical invariants govern its arithmetic: the ideal class group Cl(O_K), measuring failure of unique factorization, and the unit group O_K^×, described by the Dirichlet unit theorem. K-theory absorbs both at once.
O_K = ring of integers of a number field K, a Dedekind domain.
DEGREE 0: K_0(O_K) ~= Z (+) Cl(O_K).
The rank-Z part is the free piece; the reduced part is the
class group. So |Cl(O_K)| = h_K, the class number, lives in K_0.
DEGREE 1: K_1(O_K) ~= O_K^x (SK_1 = 0, Bass-Milnor-Serre).
By Dirichlet's unit theorem,
O_K^x ~= mu_K (+) Z^{r_1 + r_2 - 1},
roots of unity times a free part of rank r_1 + r_2 - 1.
Worked example: K = Q(sqrt(-5)), O_K = Z[sqrt(-5)].
Cl(O_K) = Z/2 (the ideal (2, 1+sqrt(-5)) is non-principal),
so K_0 = Z (+) Z/2, h_K = 2.
Units: only +-1, so K_1 = O_K^x = {+1, -1} = Z/2.Pause on the unity here. The class number and the unit rank — the two numbers that appear in the analytic class number formula for the Dedekind zeta function — are exactly the structure of K_0 and K_1. K-theory did not invent new arithmetic; it revealed that two old invariants are degree-0 and degree-1 shadows of one object.
Topology: why the letter K, why vector bundles
K-theory was born in topology. Replace ‘‘projective module over R’’ with ‘‘vector bundle over a space X’’ — the two notions match exactly when X is compact, via the Serre–Swan theorem: bundles over X are the same as projective modules over the ring C(X) of continuous functions. The Grothendieck group of vector bundles is topological K-theory K^0(X), and the letter K (from Klasse, German for class) is shared on purpose.
The conjectural bridge: K-groups and zeta values
The deepest payoff is still partly conjectural. Quillen computed the K-theory of finite fields; the higher K-groups of Z itself, K_n(Z), are far harder and link directly to special values of the Riemann zeta function ζ(s). This is the content of conjectures of Lichtenbaum and Quillen, large parts now theorems thanks to the proof of the Bloch–Kato/norm-residue conjecture by Voevodsky and Rost.
Flavour of the K(Z) <-> zeta dictionary (orders of finite groups,
for even index >= 2):
|K_{4k+2}(Z)| / |K_{4k+1}(Z)| ~ numerator/denominator of
a Bernoulli-number ratio
= essentially zeta(-1-2k).
Concrete known values:
K_0(Z) = Z
K_1(Z) = Z/2 ( = {+-1}, the units )
K_2(Z) = Z/2
K_3(Z) = Z/48
K_4(Z) = 0
K_5(Z) = Z
K_7(Z) = Z/240 <-- 240 = denominator of zeta(-3) = B_4/4 data
The appearance of 240 and 48 is NOT a coincidence: these are the
denominators of zeta values, predicted by the Lichtenbaum conjecture.Step back and look at what one construction did. Starting from nothing but ‘‘subtract isomorphism classes of modules,’’ we built K_0; pushing on automorphisms gave K_1 and the determinant; probing the relations among row operations gave K_2 and Steinberg symbols; and a single homotopy-theoretic idea produced all higher K. Along the way the class group, the unit theorem, vector bundles, and zeta values all turned out to be facets of the same object. That convergence — not any one computation — is why algebraic K-theory is worth the climb.