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An algebraic curve is a one-dimensional algebraic variety. On a smooth algebraic curve, the local ring at every closed point is a discrete valuation ring, which allows orders of zeros and poles to be recorded as a divisor on an algebraic curve.

Let $R$ be a complete discrete valuation ring, $f\in R[X]$, and $d=|f^{\prime }(a)|$ satisfy $0<d<1$ and $|f(a)|=d^2$. Suppose the second Taylor coefficient of $f$ at every point of $R$ has norm below one. One Newton iteration over a valued field step has correction $h=-f(a)/f^{\prime }(a)$ of norm $d$. Its residual has quadratic term of norm strictly below $d^2$, and all higher terms have norm at most $d^3<d^2$. The derivative changes by a term of norm below $d$, so its norm remains $d$. The strong form of Hensel lemma therefore applies from the new point, producing a root within distance $d$ of the original point. The second Taylor coefficient is $f^{\prime \prime }/2$ when that quotient is meaningful, or the division-free binomial-coefficient Taylor expression in arbitrary characteristic. A Newton sequence that reaches a root is thereafter constant; residuals decrease strictly only before that event.

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.

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.

Write a line as $L=\{x_2=x_3=0\}$ and the surface equation as $F=x_2{Q}_2+x_3{Q}_3$. Smoothness says $A=Q_2{|}_L$, $B=Q_3{|}_L$ have no common zero. Projection $[x_2:x_3]$ extends over $L$ by $[-B:A]$. Its fibres are the residual plane conics in the planes through $L$, since substituting $x_2=sw$, $x_3=tw$ gives $F=w(s{Q}_2+t{Q}_3)$. Different residual conics are disjoint everywhere, including their points on $L$, because they are fibres of one morphism of algebraic varieties. The family is flat: over the smooth base curve its local rings are torsion-free over the corresponding discrete valuation ring; smoothness of the total surface supplies the reduced-fibre conclusions.

A monic polynomial over a complete discrete valuation ring factors with prescribed coprime monic residue factors. Given monic lifts $g_r,h_r$ correct modulo ${\pi }^r$, the degree-bounded map $(u,v)\mapsto u{\bar{h}}_r+v{\bar{g}}_r$ is a bijection over the residue field, by coprimality and equality of dimensions. Solve it for the error $(f-g_r{h}_r)/{\pi }^r$, and replace the lifts by $g_r+{\pi }^{r}u,h_r+{\pi }^{r}v$. Their coefficients converge by completeness and the limiting product is $f$. The completeness assumption cannot be replaced just by localization.

A nonconstant morphism of algebraic varieties $f:C\to D$ between smooth projective integral curves is finite, and its degree is the degree of the induced extension of function fields. It is also the degree of the pullback of a rational point, with ramification multiplicities and residue degrees included: locally the finite coordinate algebra is a torsion-free module over the target discrete valuation ring, hence free of its generic rank. For $D={\mathbb{P}}^1$ and a rational function defining $f$, the pullback of infinity is its pole divisor, so the total pole degree equals $\mathrm{deg}f$. A degree-one finite map to a normal curve is an isomorphism, since a finite integral algebra inside its unchanged function field equals the integrally closed target ring.

For a discrete valuation ring with uniformizer $\pi$, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by $\pi$ and whose constant coefficient is not divisible by ${\pi }^2$. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.

For an algebraically closed field $k$ and a finitely generated field extension $K/k$ of transcendence degree one, choose $t$ with $K/k(t)$ finite. The integral closures of the two standard coordinate rings of the projective line in $K$ are finite modules and glue to a finite normalization of an algebraic curve of the projective line. A one-dimensional normal local ring is a discrete valuation ring; over the perfect field $k$ the resulting projective curve is smooth. Extension of rational maps gives uniqueness of this smooth projective model.

For a p-modular system $(K,\mathcal{O},k)$, an integral form of a finite-dimensional $KG$-module $V$ is a $G$-stable finite free $\mathcal{O}$-submodule $W\subset V$ with $K{\otimes }_{\mathcal{O}}W\cong V$. Starting with a basis lattice $L$, the sum ${\sum }_{g\in G}gL$ supplies such a form: it is finitely generated and torsion-free, hence free over the discrete valuation ring $\mathcal{O}$. Reduction gives the $kG$-module $W/\pi W$, where $\pi$ is a uniformizer.

At a point of a smooth algebraic curve, its local ring is a discrete valuation ring. A local parameter is a generator $t$ of its maximal ideal, equivalently an element of valuation one. It measures the order of a zero of a function or pole by expressing functions as powers of $t$ times units. Over the complex numbers its analytic counterpart is a local coordinate vanishing simply at the point. For an elliptic curve in a Weierstrass equation of an elliptic curve, $t=-x/y$ is the standard parameter at the identity.

At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.

Let $I$ be a complete discrete valuation ring with a continuous ${\mathbb{Z}}_p$-algebra structure. Every continuous function on ${\mathbb{Z}}_p$ with values in $I$ has a unique uniformly convergent expansion in the binomial polynomials, with $a_n={\Delta }^{n}f(0)$. This is the coefficient-ring version of the Mahler theorem. To see that $a_n\to 0$, approximate the function modulo any power of the maximal ideal by a function of period $p^k$. In the expansion of $(E-1{)}^{p^k}$, all intermediate binomial coefficients are divisible by $p$, while $(E^{p^k}-1)$ kills the periodic function. Iterating makes its differences vanish modulo the prescribed ideal power. Finite differences do not increase the uniform norm, so the same conclusion holds for the original function. Binomial inversion gives agreement at the nonnegative integers, and their density in the p-adic integers gives the expansion everywhere. This gives the dual description of p-adic measures by arbitrary bounded sequences of binomial moments, and hence the Amice transform.

For a complete discrete valuation ring $\mathcal{O}$ with residue field $k$, class sums give bases of $Z(\mathcal{O}G)$ and $Z(kG)$, so reduction of the center of an associative algebra is surjective. Idempotent lifting in this complete commutative algebra gives a bijection of primitive central idempotents, hence of blocks of a group algebra. Corresponding blocks of a group algebra have the same defect groups of a block: relative traces reduce, and a lifted trace whose reduction is the block identity is a unit in its complete block center.

For a nilpotent ideal $I$, its nilpotence index of an ideal is the least positive integer $n$ for which $I^n=0$. In $k[X]/(g^e)$, with $g$ an irreducible polynomial, the unique maximal ideal $(g)$ has index $e$. In a discrete valuation ring with $p=u{\pi }^e$, the maximal ideal of its reduction modulo $p$ has the same index. Comparing these indices under a quotient ring isomorphism recovers the ramification exponent in the Kummer-Dedekind theorem.

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.

For a profinite group $G$ and a complete discrete valuation ring $I$ with a continuous ${\mathbb{Z}}_p$-algebra structure, an $I$-valued measure is a bounded $I$-linear functional on continuous $I$-valued functions, with the uniform norm. Equivalently it assigns finitely additive values in $I$ to clopen sets. Locally constant functions are uniformly dense, so integration extends uniquely from finite sums. Countable additivity in the usual real-measure sense is not imposed. The coefficient ring may have infinite residue field and need not be compact.

For every valuation ring $D$ with fraction field $K$, consider compatible morphisms of schemesAn extension is a morphism of schemes $\ell :\mathrm{Spec}D\to X$ satisfying $\ell {|}_{\mathrm{Spec}K}=u$ and $f\ell =v$. The valuative criterion for universal closedness statesEquivalently, among quasi-compact morphisms, the existence condition characterizes universal closedness. A universally closed morphism is automatically a quasi-compact morphism; the converse direction of the criterion needs this compactness condition. All valuation rings are allowed, with arbitrary fraction fields; there is no Noetherian ring hypothesis and one cannot replace them by discrete valuation rings without additional assumptions.^[ref][ref]

A Dedekind domain is a Noetherian integrally closed domain in which every nonzero prime ideal is maximal. The equivalent local characterization is that its localizations at nonzero prime ideals are discrete valuation rings. A field is harmless as the dimension-zero case.

Write $R$ for the domain, $F$ for its fraction field and $I$ for the nonzero fractional ideal. Its proposed inverse is an $R$-submodule of $F$. Choose $0\ne d\in R$ with $dI\subseteq R$, so $d\in I^{-1}$ and the inverse is nonzero. Choose also $0\ne a\in I$. Every $x\in I^{-1}$ satisfies $xa\in R$, hence $I^{-1}\subseteq a^{-1}R$. Writing $a=r/s$ with $r,s\in R$ gives $r{I}^{-1}\subseteq sR\subseteq R$. Thus $I^{-1}$ is a fractional ideal; the Noetherian hypothesis also makes it finitely generated.

For a nonzero maximal prime ideal $\mathfrak{p}$, the discrete valuation ring $R_{\mathfrak{p}}$ makes $I_{\mathfrak{p}}$ principal, say $I_{\mathfrak{p}}=a_{\mathfrak{p}}{R}_{\mathfrak{p}}$. We need to justify that inversion commutes with this localization. One inclusion is immediate. Conversely, if $x{I}_{\mathfrak{p}}\subseteq R_{\mathfrak{p}}$, finite generation of $I$ supplies a common $s\notin \mathfrak{p}$ such that $sxI\subseteq R$. Therefore $sx\in I^{-1}$, provingIt follows that $(I{I}^{-1}{)}_{\mathfrak{p}}=R_{\mathfrak{p}}$ at every such prime ideal. Globally $I{I}^{-1}\subseteq R$. If this were a proper ideal, it would be contained in a maximal ideal and would remain proper after localizing there, a contradiction. In the field case every nonzero fractional ideal is the whole field. Hence in all casesThis proves that every nonzero fractional ideal of a Dedekind domain is an invertible fractional ideal, rather than presupposing its inverse exists.