# 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](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](source-notice.md).

The exact source excerpts [00FN-proof.tex](00FN-proof.tex) and [09H0-proof.tex](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](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/README.md), [algebra source](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/algebra.tex#L6141-L6191), [fields source](https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/fields.tex#L1167-L1202), [original Stacks project](https://stacks.math.columbia.edu/).

## 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.