feat(LocalRing/Etale): Finite étale extensions of local rings are monogenic

[bryanboehnke]

Formalization of Lemma 3.2 from https://arxiv.org/abs/2503.07846: Given a finite extension of local rings R -> S, if the extension is étale, then there exists β ∈ S such that R[β] = S. Furthermore, if f(z) ∈ R[z] is the minimal polynomial of β, then f′(β) is a unit in S. This file also includes some relevant intermediate results used in the formalization of the proof of Lemma 3.1 from the same paper.

Co-authored-by: George Peykanu gpeyka@uw.edu
Co-authored-by: Bianca Viray bviray@uw.edu
Co-authored-by: Grant Yang granty29@uw.edu

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• depends on: [Merged by Bors] - feat(RingTheory/IsAdjoinRoot): add mkOfAdjoinEqTop' #36421

[github-actions]

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[github-actions]

PR summary 9ee33d6428

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.LocalRing.Etale (new file)	2323

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Declarations diff

+ adjoin_residue_eq_top_iff_adjoin_eq_top
+ exists_adjoin_eq_top
+ finrank_eq_finrank_residueField
+ isUnit_aeval_derivative_minpoly_of_adjoin_eq_top
+ minpoly_map_residue

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
633	2	erw

Current commit a08730a1b7
Reference commit 9ee33d6428

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(RingTheory/IsAdjoinRoot): add mkOfAdjoinEqTop' #36421
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.