Discrete Valuation Rings

Valuation rings are available for the rational field and for rational function fields. For rational function fields, given an arbitrary monic irreducible polynomial $p(x) \in K[x]$, the valuation ring is

$$O_{p(x)} = \{ {f(x) \over g(x)} : f(x), g(x) \in K[x], p(x) \!\not\vert\; g(x) \}.$$

Valuations corresponding both to an irreducible element and to $\infty$ are allowed.

• Valuation ring corresponding to the discrete non-Archimedean valuation v_p of $\mbox{\bf Q}$
• Valuation ring corresponding to the discrete non-Archimedean valuation v_p of a rational function field
• Valuation ring corresponding to the valuation $v_{\infty}$ of a rational function field
• Arithmetic
• Euclidean norm, valuation
• Greatest common divisor