Ring of integers of a number field (${\mathcal{O}}_K$)

The ring of integers ${\mathcal{O}}_K$ of a number field $K$ is the ring of all algebraic integers in $K$.

Every nonzero ideal of the ring of integers of a number field factors uniquely as a finite product of nonzero prime ideals. Thus each $x\in K^{\times }$ has integer valuations ${\mathrm{ord}}_{\mathfrak{p}}(x)$, almost all zero, determined by the factorization of its principal ideal.