DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Uniformising parameters

Definition

Let $V$ be a discrete valuation ring with discrete valuation $v$ and maximal ideal $m$ . A uniformising parameter or uniformiser is an element $π \in V$ with

v (π) = 1.

This is equivalent to requiring that $π$ generate the maximal ideal $m$ . Indeed, if $v (π) = 1$ and $x \in m$ is nonzero, then $v (x) \geq 1$ , so $v (x / π) \geq 0$ and therefore $x \in (π)$ . Conversely, if $m = (π)$ and $v (π) \geq 2$ , then surjectivity of the valuation gives an element $y$ with $v (y) = 1$ , and $y \in m = (π)$ would force $1 = v (y) \geq v (π) \geq 2$ , impossible.

Depends on

Used by

Computing the length of R/(πⁿ) Example
Uniformisers and ideal arithmetic in a DVR Example
Every nonzero fraction is a unit times a power of a uniformiser Theorem
Ideals in a DVR are powers of the maximal ideal Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Mustata, Commutative Algebra, Remark 8.9 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)