Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-04
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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.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A complete discrete valuation ring is Henselian

Example

For every field k, the formal power-series ring k⟦t⟧ is a complete discrete valuation ring and hence Henselian.

Facts & Assumptions

Given: A field k and the ring k⟦t⟧.

[L1]

The ring k⟦t⟧ is a local domain with unique maximal ideal tk⟦t⟧ (For a field K, K⟦x⟧ is a domain and its nonunits form the unique maximal ideal xK⟦x⟧).

[L2]

A local domain of this form is a discrete valuation ring (Equivalent characterizations of a DVR).

[L3]

Every complete local ring is Henselian (Complete local rings are Henselian).

Verification

technique · identify the standard complete local model
1.1

By [L1], k⟦t⟧ is local with maximal ideal (t). Its t-adic topology is complete by construction of the formal power-series ring.

L1given
2.1

By [L2], this local domain is a discrete valuation ring. Applying [L3] to the complete local ring k⟦t⟧ shows that it is Henselian.

L2L3step 1.1
3.1

Therefore every complete discrete valuation ring modeled by k⟦t⟧ is Henselian.

step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources