Class number of a number field

The class number of a number field $K$ is

h_K=|\operatorname{Cl}(K)|,

the cardinality of its ideal class group. The finiteness theorem for ideal classes ensures that this is a positive integer.

Meaning of class number one

One has $h_K=1$ exactly when every ideal of $\mathcal O_K$ is principal. For example $h_{\mathbb Q}=1$ , since every ideal of $\mathbb Z$ is generated by a single integer.

A geometric use

For imaginary quadratic fields, the Bianchi cusp correspondence interprets this arithmetic number as the number of cusp ends of a hyperbolic orbifold.

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §4, “The Finiteness of the Class Number.”