Algebra prerequisites for the critical-values argument

This component contains the two proof excerpts used in AN06-U008, with a readable adaptation and explicitly marked elementary proof completions. Its source is Commutative Algebra, tag 00FN, and Fields, tag 09H0, in Unofficial Stacks Project AI Drafts, revision 565b10e987aba5969b21145a0833f42d69f96790. The inherited Stacks text is the work of the Stacks Project authors; the fork's changes and integration are AI-written drafts. The Stacks Project and its maintainers have not reviewed, approved, or endorsed the fork or this component.

The pinned source and modified source are distributed under the GNU Free Documentation License, version 1.2. The licence text is included in COPYING. This component retains that licence; it is not part of the course's CC0 grant. No new invariant sections, front-cover texts, or back-cover texts are designated by this component. The original fork's licence statement is preserved in Source notice.

The exact source excerpts 00FN-proof.tex and 09H0-proof.tex are unchanged. The readable Markdown adapts notation and supplies elementary prerequisites rather than silently strengthening the source proof. Modifications on 2026-10-03: OpenAI Codex, GPT-6.1 Sol, Ultra effort; complete ACC equivalence, the monotone-subsequence argument, uniform two-parameter stabilization, quotient/localization arguments, polynomial division and Bézout, and the characteristic-zero root consequence. These are ordinary mathematical written arguments, not proof-assistant certification or independent human review.

Source/history links: fork README and licence statement, algebra source, fields source, original Stacks project.

History

The source edition is Unofficial Stacks Project AI Drafts, revision 565b10e987aba5969b21145a0833f42d69f96790, incorporating the Stacks Project authors’ mathematical text and the fork’s AI draft modifications; the source and history links above identify that edition. The AN-06 component was adapted by GPT-6.1 Sol (OpenAI), Ultra, on 3 October 2026, with the completions listed above. On 5 October 2026, GPT-6 Astra (OpenAI), Ultra, checked both complete readers and added explicit quotient/localization constructions, finite algebraic extensions, the prescribed-element transcendence-basis argument, and rational/algebraic derivation extensions. The two source excerpts and COPYING remain unchanged. These modified readers retain GFDL 1.2. No independent human review or Stacks Project endorsement is asserted.