Discrete valuation ring

A discrete valuation ring is a principal ideal domain with exactly one nonzero maximal ideal. Every nonzero element of its fraction field is a unit times a unique integer power of a uniformizer.

A field with a surjective discrete valuation, complete for the associated non-Archimedean metric. Its valuation ring is a discrete valuation ring, and its maximal ideal is generated by a uniformizer. Completeness allows successive congruence solutions to converge to actual roots, which underlies the Hensel lemma. Non-Archimedean local fields are examples; completeness alone does not require the residue field to be finite.

For a normalized discrete valuation, a uniformizer $\pi$, and a chosen set $A$ of representatives of the residue field, every existing nonzero element has a unique displayed expansion, with its first digit not in the maximal ideal. Starting from $y_0={\pi }^{-v(x)}x$, choose $a_j$ representing $y_j$ modulo the maximal ideal and put $y_{j+1}=(y_j-a_j)/\pi$. The error after $r$ digits has valuation at least $v(x)+r$, so the sums converge to $x$ even without completeness. Conversely arbitrary digit sequences need not converge in an incomplete field: in $\mathbb{Q}$ with its three-adic valuation, the digits of $\sqrt{10}\in {\mathbb{Q}}_3$ give a counterexample. Hensel lemma gives this square root, but it is not rational.

Let $R$ be a one-dimensional normal Noetherian local domain. Choose $0\ne x$ in its maximal ideal $\mathfrak{m}$ and a minimal $n$ with ${\mathfrak{m}}^n\subseteq (x)$. If $n>1$, choose $y\in {\mathfrak{m}}^{n-1}\setminus (x)$ and put $z=y/x$. Then $z\notin R$ but $z\mathfrak{m}\subseteq R$. If $z\mathfrak{m}\subseteq \mathfrak{m}$, the determinant trick makes $z$ integral, contrary to normality. Thus $zu$ is a unit for some $u\in \mathfrak{m}$, and $v=u(zv)/(zu)$ for every $v\in \mathfrak{m}$ proves $\mathfrak{m}=(u)$. The case $n=1$ is already principal. Such a local domain is a discrete valuation ring.

A complete discrete valuation ring is a discrete valuation ring complete for its maximal-ideal-adic topology. Equivalently, the canonical map to the inverse limit of the residue rings is an isomorphism: compatible residue classes determine a Cauchy sequence, and completeness gives its unique limit. Standard examples are ${\mathbb{Z}}_p$ and $k[[t]]$. This completeness is the hypothesis that makes Newton iteration over a valued field converge in the ring in the usual Hensel lemma.

A discrete valuation on a field $K$ is a surjective group homomorphism $v:K^{\times }\to \mathbb{Z}$ satisfying $v(x+y)\ge \min (v(x),v(y))$ whenever $x+y\ne 0$. Its valuation ring is $\{0\}\cup \{x:v(x)\ge 0\}$.

A discretely valued field is a field equipped with a discrete valuation. It is complete when every Cauchy sequence for the induced metric converges.

The value group of a valued field is the ordered abelian group formed by the values of its nonzero elements. A valuation is discrete when its value group is infinite cyclic with the order inherited from $\mathbb{Z}$.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.