Lang — Algebra (Fast Track 3.01) — Audit + Gap Plan

Book: Serge Lang, Algebra, revised third edition. Graduate Texts in Mathematics 211, Springer 2002 (corrected printing of 1965/1971/1993 Addison-Wesley editions). ISBN 978-0-387-95385-4. Approx. xv + 914 pp.

Fast Track entry: 3.01 — the first book in §3 "Modern Geometry, Algebraic Topology, Mathematical Foundations." Lang is the canonical graduate-algebra anchor of the Fast Track and the prerequisite spine for the rest of §3: 3.02 Gel'fand–Manin Homological Algebra, 3.10 Atiyah K Theory, 3.11 Fulton–Harris Representation Theory, 3.12 Serre Complex Semisimple Lie Algebras, 3.13 Serre Lie Algebras and Lie Groups, 3.21 Hartshorne Algebraic Geometry, 3.22 Griffiths–Harris, 3.34 Manin Modern Number Theory. The Fast Track source page lists Lang's TOC under the §3 toggle with no body prose — the book is treated as a reference floor that downstream §3 books assume in toto.

Purpose of this plan: P1 audit-and-gap pass. Output is a concrete punch-list of new units to write and existing units to deepen so that Algebra is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

The audit surface here is the largest single-volume gap in the corpus. The Codex 01-foundations/02-algebra/ directory currently contains exactly one shipped unit (01.02.01-group.md, Wave-3 Phase-3.2, shipped 2026-05-07), against Lang's ~30 chapters spanning groups, rings, modules, fields, Galois theory, semisimple representations, commutative algebra, and homological algebra. The campaign brief framed the foundation as "10 shipped units"; that count is aspirational — on-disk it is one. Several Lang topics have shipped piecemeal under neighbouring directories: linear algebra units (10 in 01-foundations/01- linear-algebra/), representation theory units (26 in 07-representation- theory/), and commutative-algebra-flavoured units in 04-algebraic-geometry/02-schemes/. Closing the Lang gap therefore requires not just new 01.02.* units but rigorous cross-referencing to the existing rep-theory and alg-geom corpus so that core theorems (Sylow, Galois, Wedderburn, Schur, Maschke, Nakayama, Hilbert basis, Noether normalisation, Nullstellensatz) are not re-derived inconsistently.

This pass is intentionally focused on Parts I–III of Lang (Groups, Rings, Modules, Fields/Galois, Linear/Semisimple). Part IV (Homological algebra Ch. XX–XXI) is deferred to the 3.02 Gel'fand–Manin audit — the two books cover the same ground and the Codex should treat Gel'fand– Manin as the primary homological-algebra reference, not Lang's appendix. Lang's commutative-algebra material in Ch. IX–X (Noetherian rings, integral extensions, Krull dimension, localisation) overlaps the Atiyah–Macdonald appendix to Hartshorne and is partially deferred to the 3.21 Hartshorne audit — Codex commitments are recorded here but production credit lives in the Hartshorne plan.

§1 What Lang's Algebra is for

Lang is the canonical graduate-level abstract algebra textbook in the American tradition. Where Dummit–Foote Abstract Algebra and Aluffi Algebra: Chapter 0 are graduate-introductory and pedagogically patient, and where Bourbaki Algèbre and Atiyah–Macdonald Introduction to Commutative Algebra are encyclopaedic-systematic and minimally- expository respectively, Lang is terse, fast, abstract, and very wide. Lang's editorial bet: a graduate student who has worked through Dummit–Foote or Artin at the undergraduate level should be able to absorb the entire abstract-algebra graduate curriculum from one source, with appendices and exercises supplying the connections to fields the student will later need (algebraic geometry, number theory, Lie theory, representation theory). The book is universally adopted as the qualifying- exam reference for U.S. graduate algebra.

Distinctive editorial choices, in roughly the order Lang develops them:

Categories from page one. Lang opens Ch. I with a brief "categories and functors" section before defining groups, then uses functorial language throughout (free objects as left-adjoints, universal properties as the definition of tensor products, etc.). This is the principal point of contrast with Dummit–Foote (which defers categorical language to a late chapter) and with Aluffi (which builds categories first in a more leisurely way). Codex implication: Lang notation crosswalk must be explicit about when a "universal property" is in play, because the Codex follows Aluffi's slower category-first style in foundations units but graduate § targets adopt Lang's terse functorial framing.
Groups, rings, modules in parallel (Ch. I–III). Lang sets up parallel structural theorems (homomorphism, isomorphism, correspondence theorems) across the three categories, then treats finite group theory (Sylow, simplicity of $A_{n}$ , solvable and nilpotent groups, Jordan–Hölder) in concentrated form in Ch. I §6–9. Rings (Ch. II) and modules (Ch. III) are treated before polynomial rings (Ch. IV), so that polynomial-ring theory is presented as the canonical example of the general module theory, not the other way around.
Fields and Galois theory (Ch. V–VIII) treated abstractly. Lang develops algebraic and transcendental extensions, separability, normality, Galois correspondence, Kummer theory, and infinite Galois theory (with the Krull topology and profinite groups) as one integrated story. Distinctive: infinite Galois theory and profinite groups are introduced before the reader leaves Part II, where most undergraduate texts defer them or omit them.
Commutative algebra (Ch. IX–X) as algebraic-geometry preparation. Localisation, Noetherian rings (Hilbert basis theorem, Krull's intersection theorem, Krull dimension), integral extensions (Nakayama, going-up / going-down, Noether normalisation, Nullstellensatz), and completions. Lang's coverage here is the bridge to Hartshorne's Ch. I; Lang's notation is the one downstream Hartshorne units must adopt.
Real fields, absolute values, valuation rings (Ch. XI–XII). Orderings on fields, real closures, places and valuations, completions of valued fields, the local–global perspective with Ostrowski's theorem. Bridge to algebraic number theory (Manin 3.34) and $p$ -adic Lie theory (Serre 3.13).
Linear algebra and bilinear forms (Ch. XIII–XV) at the abstract level. Matrices and linear maps over a commutative ring, structure theorem for modules over a PID (with the rational and Jordan canonical forms as corollaries), bilinear and quadratic forms, Witt's theorem. The Codex has 10 shipped linear-algebra units following Shilov; Lang's versions are the generalised-ring versions of those, and the audit must record which units need a graduate-tier deepening to absorb Lang's coefficient-ring generality.
Representations and group rings (Ch. XVII–XVIII). Semisimple rings and modules (Wedderburn–Artin structure theorem), the Jacobson radical, representations of finite groups via group algebras ( $k [G]$ -modules), Maschke's theorem, character theory, induced representations, Frobenius reciprocity (in the algebraic presentation). The Codex has 26 shipped representation-theory units (mostly following Fulton–Harris); Lang's version is the algebra-side account that downstream rep-theory work depends on (Wedderburn first, characters second).
Appendices and homological-algebra coda (Ch. XIX–XXI). Tensor products of modules redone for noncommutative coefficients, general homology theory (chain complexes, derived functors, $Ext$ and $Tor$ ), finite free resolutions. Codex defers most of this to 3.02 Gel'fand–Manin; only the Lang-specific tensor-product notation and the bare definitions of $Ext / Tor$ are on this audit's scope.

Cited peer sources for §1 framing (used throughout this plan):

David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd ed., Wiley 2004. The dominant U.S. graduate-introductory algebra text; pedagogically patient where Lang is terse. Used here as the "slow reference" for first-pass exposition of Sylow, Galois, modules over PIDs. [ref: TODO_REF Dummit-Foote 2004]
Paolo Aluffi, Algebra: Chapter 0, AMS Graduate Studies in Mathematics 104, 2009. Category-first reorganisation of the same material; the closest single text to the Codex's own pedagogical style. Used here for cross-checking notation when Lang's categorical framing is too compressed. [ref: TODO_REF Aluffi 2009]
Nicolas Bourbaki, Algèbre, Chapters 1–10, Hermann then Springer (various dates 1942–2007; English translation Springer 1989). The encyclopaedic-systematic comparison reference; consulted for precise statements where Lang is loose (especially tensor products, semisimple modules, structure of finite-dimensional algebras). [ref: TODO_REF Bourbaki Algèbre]
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley 1969 (reprinted Westview/CRC 1994). The compact commutative-algebra reference; Lang Ch. IX–X covers the same material in a different order. Used to anchor the Codex's commutative-algebra prep for Hartshorne. [ref: TODO_REF Atiyah-Macdonald 1969]

The Codex equivalence target inherits Lang's bet: graduate-tier units should be terse, functorial, and self-consistent across the three parallel theories (groups, rings, modules). A reader who has worked through the Codex 01.02.* and 01.01.* strands should be able to open Hartshorne Ch. I, Fulton–Harris §1, or Serre Lie Algebras and Lie Groups Part I and recognise the territory without further preparation.

§2 Coverage table (Codex vs Lang)

Cross-referenced against the current ~313-unit corpus. ✓ = covered, △ = partial / different framing or shipped under a neighbouring directory, ✗ = not covered. Lang chapter numbers refer to the GTM 211 third edition.

Lang topic	Lang loc.	Codex unit(s)	Status	Note
Categories and functors	Ch. I §11 (and appendix)	—	✗	Gap. Codex has implicit category use but no foundational category-theory unit; deferred to a separate `01.02.0X` category-theory stub (see §3 P-1).
Group, subgroup, homomorphism	Ch. I §1–3	`01.02.01-group.md`	✓	Shipped. Beginner/Intermediate/Master. Master tier cites Lang §I anchor.
Cosets, quotient group, normal subgroup, isomorphism theorems	Ch. I §3–4	—	✗	Gap. Foundational.
Group action, orbit-stabiliser, conjugation	Ch. I §5	△	△	`03.03.02-group-action.md` (per `01.02.01` successors list) covers Lie-group actions; algebraic group-action unit needed.
Sylow theorems	Ch. I §6	—	✗	Gap (high priority). Load-bearing for rep theory and number theory.
Free groups, free abelian groups, presentations	Ch. I §12	—	✗	Gap. Needed for fundamental-group computations (cross-Brown 1.05).
Solvable, nilpotent groups; Jordan–Hölder	Ch. I §9–10	—	✗	Gap. Needed for Galois solvability.
Symmetric, alternating, simplicity of $A_{n}$	Ch. I §5	△	△	Touched in `07.05.*` symmetric-group rep-theory units; structural unit (simplicity of $A_{n}$ , conjugacy classes) absent.
Ring, ideal, quotient ring, homomorphism	Ch. II §1–3	—	✗	Gap. Foundational.
Polynomial ring, UFD, PID, ED	Ch. II §3–4; Ch. IV	—	✗	Gap. Foundational.
Localisation of rings	Ch. II §3	△	△	Implicit in `04.02.02-affine-scheme.md`; no standalone algebraic unit.
Chinese Remainder Theorem	Ch. II §2	—	✗	Gap. Small unit.
Module, submodule, quotient, hom	Ch. III §1–4	—	✗	Gap. Foundational.
Direct sum, direct product, free module	Ch. III §4–6	△	△	Vector-space versions in `01.01.03-vector-space.md`; module version absent.
Tensor product of modules	Ch. III §3 (and Ch. XVI)	△	△	Rep-theoretic version in `07.01.06-tensor-product-of-representations.md`; algebraic-module version absent.
Exact sequence, snake lemma	Ch. III §9	—	✗	Gap. Defer machinery to 3.02 Gel'fand–Manin; need a small stub here.
Polynomial ring properties: Gauss's lemma, Eisenstein, content	Ch. IV §1–4	—	✗	Gap.
Field, characteristic, prime field	Ch. V §1	△	△	`01.01.01-field.md` covers Beginner/Intermediate; Lang-tier characteristic-theory deepening needed.
Algebraic extension, degree	Ch. V §1–2	—	✗	Gap.
Splitting field, normal extension	Ch. V §3–4	—	✗	Gap.
Separable extension, perfect field	Ch. V §6	—	✗	Gap.
Galois group, Galois correspondence (finite)	Ch. VI §1–2	—	✗	Gap (high priority). Originator: Galois 1832.
Cyclotomic extensions, roots of unity	Ch. VI §3	—	✗	Gap.
Solvability by radicals, insolubility of quintic	Ch. VI §6–7	—	✗	Gap. Signature theorem of Part II.
Kummer theory, Artin–Schreier	Ch. VI §8	—	✗	Gap (Master tier).
Infinite Galois theory, Krull topology, profinite groups	Ch. VII	—	✗	Gap (Master tier).
Transcendental extensions, transcendence basis	Ch. VIII	—	✗	Gap. Needed for Hartshorne dimension theory.
Noetherian ring, Hilbert basis theorem	Ch. X §1	△	△	Hilbert basis is invoked in `04.02.07-nullstellensatz-and-dimension-theory.md`; no standalone proof.
Integral extension, going-up / going-down	Ch. VII §1–2 (and Ch. IX)	—	✗	Gap. Atiyah–Macdonald §5 parallel; partially deferred to Hartshorne audit.
Nullstellensatz	Ch. IX §1–2	△	△	`04.02.07-nullstellensatz-and-dimension-theory.md` covers the geometric statement; pure algebra proof absent.
Noether normalisation	Ch. IX §1	—	✗	Gap. Algebraic proof needed; cross-ref Hartshorne.
Krull dimension, Krull's principal ideal theorem	Ch. X §3	△	△	Touched in `04.02.07`; structural unit absent.
Nakayama's lemma	Ch. X §4	—	✗	Gap. Small but load-bearing for Hartshorne and Atiyah K-theory.
Completion, $I$ -adic topology	Ch. X §5	—	✗	Gap. Deferred from this audit to a 3.34 Manin number-theory stub.
Ordered field, real closure	Ch. XI	—	✗	Gap (low priority, Master tier).
Absolute values, places, Ostrowski	Ch. XII	—	✗	Gap (Master tier). Cross-ref Manin 3.34.
Linear maps over a ring, matrices	Ch. XIII	△	△	Vector-space version in `01.01.*` (10 units); Lang's coefficient-ring generality deepening needed on `01.01.05-linear-transformation-rank-nullity.md`.
Determinants (over commutative rings)	Ch. XIII §4	△	△	`01.01.07-determinant.md` is field-coefficient; deepening to commutative-ring coefficients needed.
Structure theorem for modules over a PID	Ch. III §7 / Ch. XIV	—	✗	Gap (high priority). Rational and Jordan canonical form follow as corollaries; existing `01.01.11-jordan-canonical-form.md` should be reframed as a corollary, not a standalone construction.
Bilinear, sesquilinear, quadratic forms; Witt's theorem	Ch. XV	△	△	`01.01.15-bilinear-quadratic-form.md` exists; Witt cancellation/decomposition absent.
Tensor algebra, exterior algebra, symmetric algebra	Ch. XVI	△	△	Used implicitly in `04.08.01-sheaf-of-differentials.md`; standalone units absent.
Semisimple ring, semisimple module	Ch. XVII §1–3	—	✗	Gap (high priority).
Wedderburn–Artin structure theorem	Ch. XVII §4	—	✗	Gap (high priority). Originator: Wedderburn 1908, Artin 1927.
Jacobson radical	Ch. XVII §6	—	✗	Gap.
Maschke's theorem	Ch. XVIII §1	✓	✓	`07.02.01-maschke-theorem.md` shipped. Cross-link from new Lang unit.
Group algebra $k [G]$ , regular representation	Ch. XVIII §1–2	△	△	`07.01.05-regular-representation.md` covers rep-theoretic angle; algebra-side framing absent.
Schur's lemma	Ch. XVII §1	✓	✓	`07.01.02-schur-lemma.md` shipped. Cross-link.
Characters and orthogonality	Ch. XVIII §5	✓	✓	`07.01.03-character.md`, `07.01.04-character-orthogonality.md` shipped. Cross-link.
Induced representations, Frobenius reciprocity	Ch. XVIII §7	✓	✓	`07.01.07-induced-representation.md`, `07.01.08-frobenius-reciprocity.md` shipped. Cross-link; Lang's algebra-side proof should be referenced.
General homology theory	Ch. XX	—	✗	Deferred to 3.02 Gel'fand–Manin audit.
Finite free resolutions, $Ext$ , $Tor$	Ch. XXI	△	△	`04.03.06-derived-functors-and-ext.md` exists for the sheaf-theoretic angle; pure-algebra version deferred to 3.02.

Aggregate coverage estimate (Parts I–III only, the in-scope chapters): ~12% of Lang's named theorems in Ch. I–XVIII have a Codex unit that covers them at any tier. Of those, half are △ (covered under a neighbouring directory with a different framing — e.g., Maschke under rep-theory uses representation-language exclusively, not algebra-side Wedderburn-decomposition language). True ✓ count: ~8 of Lang's ~80 named theorems in scope.

Closing the gap to ≥95% effective coverage requires ~25 new units plus ~15 deepening passes on existing units (see §3).

§3 Gap punch-list (P3 — priority-ordered)

Items numbered with provisional unit IDs in 01.02.* (foundations algebra). Some items will cross-list into 01.03.* (a new commutative-algebra subdirectory to be opened) once the Hartshorne audit confirms its scope.

Priority 0 — blockers (must ship before downstream §3 work continues)

The following four downstream books currently sit on Lang's prerequisite shelf with no enforced gate: 3.11 Fulton–Harris (rep-theory units in 07-* already produced), 3.21 Hartshorne (alg-geom units in 04-* already produced), 3.13 Serre Lie groups, 3.34 Manin number theory. Each of these has shipped Codex units that implicitly cite Lang theorems not yet covered in the Codex. Continuity gate: the P-1 Lang units below should ship before further 04.* or 07.* units are added, or those new units must declare pending_prereqs: true.

Priority 1 — high-leverage, captures Lang's central content

01.02.02 Subgroup, coset, quotient group, isomorphism theorems. Foundational. ~1500 words, three-tier. Dummit-Foote §3 anchor; Lang §I.3 anchor.
01.02.03 Group action, orbit-stabiliser, class equation. Algebraic framing distinct from 03.03.02-group-action.md (which targets Lie groups). Three-tier; Master tier covers Burnside.
01.02.04 Sylow theorems. Originator: Sylow 1872. The load-bearing finite-group-theory result. Three-tier. Master tier includes worked classification of groups of order $\leq 60$ . Cross- linked from 07.05.* symmetric-group units.
01.02.05 Solvable group, nilpotent group, Jordan–Hölder. Needed for Galois solvability by radicals (P-1 item 12 below).
01.02.06 Ring, ideal, quotient ring; isomorphism theorems for rings. Foundational; ~2000 words. Cite Dummit-Foote §7–8 anchor, Aluffi §III anchor.
01.02.07 Polynomial ring; UFD, PID, ED; Gauss's lemma; Eisenstein. Foundational. Cross-link to existing 01.01.01-field.md and to 04.02.* affine-scheme units.
01.02.08 Localisation of a commutative ring. Algebraic framing; cross-link to 04.02.02-affine-scheme.md. Atiyah–Macdonald §3 anchor.
01.02.09 Module, submodule, quotient module, free module, direct sum. Foundational. ~1800 words.
01.02.10 Tensor product of modules (commutative case). Lang Ch. III §3 + Ch. XVI anchor. Universal-property definition; explicit construction. Cross-link to 07.01.06-tensor-product-of- representations.md (which becomes a corollary). Master tier covers the noncommutative case from Ch. XVI §5.
01.02.11 Exact sequence, short five lemma, snake lemma (stub). Small unit (~1000 words). Pure-algebra version; full homological machinery deferred to 3.02 Gel'fand–Manin.
01.02.12 Algebraic field extension, degree, splitting field. Foundational for Galois. Three-tier.
01.02.13 Separability, normality, Galois extension; Galois correspondence (finite). Originator: Galois 1832 (manuscripts; posthumous 1846 J. de Math. Pures Appl., ed. Liouville). The signature theorem of Part II. Three-tier; Master tier includes insolubility of the quintic as worked example. ~3500 words — largest single unit in this punch-list.
01.02.14 Structure theorem for finitely generated modules over a PID. Lang Ch. III §7 anchor. Three-tier. Existing 01.01.11-jordan-canonical-form.md must be deepened (Master-tier section) to derive JCF and rational canonical form as corollaries; this is a deepening, not a new unit.
01.02.15 Semisimple module, semisimple ring; Wedderburn–Artin structure theorem. Originator: J. H. M. Wedderburn 1908 (Proc. London Math. Soc. (2) 6, 77–118, on hypercomplex number systems); Artin 1927 generalisation. Three-tier; Master tier covers the Jacobson radical (Ch. XVII §6) in a §Bonus. Cross-link to 07.02.01-maschke-theorem.md and to 07.01.02-schur-lemma.md, both of which become corollaries of Wedderburn–Artin in the Lang-style algebra-side framing.
01.02.16 Nakayama's lemma. Small unit (~1000 words) but load-bearing for Hartshorne and Atiyah K-theory. Originator: Krull (1938) / Azumaya (1951) / Nakayama (1951) — attribution contested; Lang names it after Nakayama. Notation crosswalk paragraph required.
01.02.17 Hilbert basis theorem; Noetherian rings and modules. Originator: Hilbert 1890 Math. Ann. 36. Three-tier. Cross-link to 04.02.07-nullstellensatz-and-dimension-theory.md which currently invokes Hilbert basis without a Codex proof.
01.02.18 Integral extension; Noether normalisation; weak Nullstellensatz. Originator: Noether 1926 Math. Ann. 96 (algebraic-set normalisation); 1927 (general integral closure). Three-tier; Master tier proves strong Nullstellensatz from weak + Noether normalisation. Cross-link reframes the geometric Nullstellensatz unit in 04.02.07 as the geometric corollary.

Priority 2 — important deepenings and Master-tier extensions

01.02.19 Tensor algebra, exterior algebra, symmetric algebra. Lang Ch. XVI anchor. Cross-link from 04.08.01-sheaf-of- differentials.md (currently uses exterior powers without a Codex definition).
01.02.20 Free group, free product, group presentation. Lang Ch. I §12. Cross-link from Brown 1.05 fundamental-group punch-list.
01.02.21 Transcendental extension, transcendence basis, transcendence degree. Needed for Hartshorne dimension theory.
01.02.22 Krull dimension, Krull's principal ideal theorem. Cross-link to 04.02.07. Partially deferred to Hartshorne audit; record commitment here.
01.02.23 Group algebra $k [G]$ , regular representation (algebra-side framing). Bridge unit between Lang Ch. XVIII and 07.01.05-regular-representation.md. Reframes the latter under Wedderburn–Artin decomposition.
Deepenings on existing units (no new ID):
- 01.01.01-field.md — add Master-tier section on prime field, characteristic, Frobenius.
- 01.01.05-linear-transformation-rank-nullity.md — Master-tier coefficient-ring generality (linear maps over a commutative ring, not just a field).
- 01.01.07-determinant.md — Master-tier deepening to commutative coefficients; statement of the Cayley–Hamilton via "determinant trick" (Atiyah–Macdonald §2 anchor).
- 01.01.11-jordan-canonical-form.md — Master-tier reframing as corollary of 01.02.14 PID structure theorem.
- 01.01.15-bilinear-quadratic-form.md — Master-tier Witt cancellation and Witt decomposition.
- 01.02.01-group.md — add §Notation paragraph crosswalking Lang's categorical framing (Ch. I §11) with the Codex's slower exposition.

Priority 3 — Master-tier survey pointers (low priority)

01.02.24 Cyclotomic extensions, roots of unity, Artin–Schreier extensions. Master-only.
01.02.25 Infinite Galois theory, Krull topology, profinite groups. Master-only; pointer to Lang Ch. VII and Serre Cohomologie Galoisienne for follow-up.
01.02.26 Ordered field, real closure; absolute values, Ostrowski's theorem. Master-only. Pointer toward Manin 3.34.

Priority 4 — deferred to neighbouring audits

Homological algebra (Lang Ch. XX–XXI). Defer to 3.02 Gel'fand– Manin audit. Codex commitment: when the 3.02 audit ships, this plan is amended to record cross-references.
Completions and $I$ -adic topology (Lang Ch. X §5). Defer to 3.34 Manin number-theory audit.
Commutative algebra detail beyond Nakayama / Noether normalisation / Hilbert basis (Lang Ch. IX–X residue). Defer to 3.21 Hartshorne audit's Atiyah–Macdonald appendix.

§4 Implementation sketch (P3 → P4)

For a full Lang Parts I–III coverage pass, items 1–17 are the minimum set. Realistic production estimate (calibrated against the Bott-Tu, Lawson-Michelsohn, and Brown 1.05 batches):

~3 hours per standard unit, ~5 hours for the large units (Galois correspondence #12, Wedderburn–Artin #14, PID structure theorem #13).
Priority-1 unit count: 17 units, of which 3 are large. $14 \times 3 + 3 \times 5 = 57$ hours. Plus ~6 deepening passes averaging ~1.5 hours each = ~9 hours. Total ~66 hours of focused production for P-1.
Priority-2 unit count: 5 units + 6 explicit deepenings = ~15 hours.
Priority-3 unit count: 3 Master-only units = ~6 hours.
Grand total to ≥95% Lang equivalence (Parts I–III, in scope): ~87 hours. Fits a 10–12 day focused window. Larger than any previous single-book audit's production estimate; reflects Lang's role as the foundation of §3.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, the following P-1 units should carry originator-prose treatment:

Item 4 (Sylow) — L. Sylow, "Théorèmes sur les groupes de substitutions," Math. Ann. 5 (1872), 584–594.
Item 12 (Galois correspondence) — É. Galois, "Mémoire sur les conditions de résolubilité des équations par radicaux," 1831 manuscript, published posthumously by J. Liouville in J. de Math. Pures Appl. (1) 11 (1846), 381–444. The originating document of group theory and Galois theory together; the unit should treat the modern Galois correspondence as the Artin–Lang reformulation of Galois's original results.
Item 14 (Wedderburn–Artin) — J. H. M. Wedderburn, "On hypercomplex numbers," Proc. London Math. Soc. (2) 6 (1908), 77–118; E. Artin, "Zur Theorie der hyperkomplexen Zahlen," Abh. Math. Sem. Univ. Hamburg 5 (1927), 251–260.
Item 16 (Nakayama) — W. Krull, "Dimensionstheorie in Stellenringen," J. Reine Angew. Math. 179 (1938); G. Azumaya, "On maximally central algebras," Nagoya Math. J. 2 (1951); T. Nakayama, "A remark on finitely generated modules," Nagoya Math. J. 3 (1951). The attribution-contested origin should be documented.
Item 17 (Hilbert basis) — D. Hilbert, "Ueber die Theorie der algebraischen Formen," Math. Ann. 36 (1890), 473–534.
Item 18 (Noether normalisation) — E. Noether, "Idealtheorie in Ringbereichen," Math. Ann. 83 (1921), 24–66 (the foundational Noetherian-ring paper); E. Noether, "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern," Math. Ann. 96 (1927), 26–61 (the integral-closure / normalisation paper).
The "modern consolidation" originator-citation for the organisation of the whole material (Parts I–III taken as a single graduate course) is Serge Lang himself: the first edition of Algebra (Addison-Wesley 1965) set the template that Dummit–Foote, Hungerford, Aluffi, and every subsequent graduate algebra text reorganises in response to.

Notation crosswalk. Lang's distinctive notational choices that the Codex must record (in docs/specs/UNIT_SPEC.md §11 and as §Notation paragraphs in the affected units):

Lang writes $A^{*}$ for the group of units of a ring $A$ ; Codex convention TBD — the rep-theory units use $A^{\times}$ . Standardise on $A^{\times}$ project-wide; add notation note to 01.02.06.
Lang's $M \otimes_{A} N$ tensor product is over a commutative ring unless noted otherwise; noncommutative case in Ch. XVI uses $M \otimes_{A}^{R} N$ . Codex follows the same convention; record in 01.02.10.
Lang's $Gal (K / k)$ uses the quotient notation (extension on top); Codex follows. Many number-theory references write $Gal (K ∣ k)$ ; flag in 01.02.13 notation paragraph.
Lang's "semisimple" allows a zero module; Bourbaki's does not. Lang's convention adopted; record in 01.02.15.
Lang's radical of a ring is the Jacobson radical; the nilradical is distinguished as "nilradical." Codex follows; record in 01.02.15 and 01.02.17.

§5 What this plan does NOT cover

Homological algebra in depth (Lang Ch. XX–XXI). Deferred to the 3.02 Gel'fand–Manin Homological Algebra (Algebra V) audit. Only the bare snake-lemma stub (item 11) sits inside the Lang audit.
Commutative-algebra detail beyond Nakayama, Noether normalisation, Hilbert basis, integral extensions, Krull dimension. Deferred to the 3.21 Hartshorne audit's Atiyah–Macdonald appendix scope. In particular: associated primes, primary decomposition, depth and Cohen–Macaulay, regular local rings, completions, Hensel's lemma. Codex commitments are recorded in this plan's Priority-4 section but production credit lives in the Hartshorne audit.
Field-theoretic specialisation to number theory. Real closures (Ch. XI), absolute values and Ostrowski (Ch. XII) are Master-only pointer units (item 26) in this plan; full treatment defers to a 3.34 Manin number-theory audit when that book is added to the campaign.
Exercise pack. Lang's exercises are famously challenging and many are open-ended research-pointer problems. Exercise pack production is a P3-priority-3 follow-up after the priority-1 units ship.
Line-number inventory of every named theorem. ~900-page book; the audit here works from chapter structure and named-theorem level. A full P1 line-number audit would refine exercise-pack scope but would not change the punch-list.

§6 Acceptance criteria for FT equivalence (Lang Parts I–III)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

All 17 Priority-1 units have shipped at three-tier (Beginner / Intermediate / Master) with tier_anchors citing Lang's § numbers in the Master tier.
Existing 01.01.* units are deepened per §3 Priority-2 item 23.
Existing 07.* rep-theory units are cross-linked from 01.02.15 Wedderburn–Artin so that Maschke / Schur / character orthogonality / induced reps / Frobenius reciprocity all reference the algebra-side framing.
Existing 04.02.07-nullstellensatz-and-dimension-theory.md is cross-linked from 01.02.17 (Hilbert basis) and 01.02.18 (Noether normalisation).
≥95% of Lang's named theorems in Ch. I–XVIII map to Codex units (currently ~12%; after Priority-1 units this rises to ~75%; after Priority-1 + 2 to ~92%; full ≥95% requires Priority-3 selective pickups + the rep-theory cross-links from item 22).
≥90% of Lang's worked computations in Ch. I–XVIII have a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded per §4 (six items).
Pass-W weaving connects the new 01.02.* units to the existing 01.01.* linear-algebra strand, the 04.02.* algebraic-geometry schemes strand, and the 07.* representation-theory strand via lateral connections; each new unit has at least 2 outbound and 2 inbound connections recorded in manifests/production/connections.json.

The 17 Priority-1 units close most of the equivalence gap. Priority-2 closes the linear-algebra / rep-theory cross-linkage. Priority-3 + 4 are deepenings and deferrals.

§7 Sourcing

Status. BUY (per docs/catalogs/FASTTRACK_BOOKLIST.md row 3.01). Springer GTM 211 retail; no free author PDF.
Local copy. Not present in reference/textbooks-extra/ as of this audit (2026-05-18). Only Lang's Basic Mathematics is on disk. Action: acquire and add to reference/fasttrack-texts/03-modern- geometry/ as Lang-Algebra.pdf to mirror the pattern of other §3 texts. audit_completeness: full — this audit works from knowledge of Lang's well-documented TOC (verifiable against Springer publisher page, Wikipedia summary, and the Fast Track source page's TOC images at reference/fast-track/images/image-33__c7d6862489.png through image-37__9d46db4177.png); the punch-list does not depend on line-level inspection.
Peer references for cross-checking and notation crosswalk (all TODO_REF until physical copies are added to reference/):
- Dummit, Foote, Abstract Algebra, 3e (Wiley 2004). Pedagogically slower; primary cross-check for Sylow, Galois, modules-over-PID.
- Aluffi, Algebra: Chapter 0 (AMS 2009). Category-first reorganisation; closest to Codex style.
- Bourbaki, Algèbre Ch. 1–10 (Springer English ed. 1989). Encyclopaedic precision; consulted where Lang is loose.
- Atiyah, Macdonald, Introduction to Commutative Algebra (Addison- Wesley 1969). The compact commutative-algebra reference; Lang Ch. IX–X parallel.
Originator papers. See §4 originator-prose targets list (Sylow 1872, Hilbert 1890, Wedderburn 1908, Galois 1846 posth., Noether 1921 + 1927, Artin 1927, Krull 1938, Azumaya / Nakayama 1951).
License note. Lang's Algebra is in print, in copyright, commercial; cite excerpts under fair-use for educational annotation only. Do not redistribute PDF.

Audit completeness: full (Parts I–III, the in-scope chapters). Parts IV (homological), Ch. XI–XII (real / valued fields), and Ch. IX–X residue commutative algebra are explicit deferrals to neighbouring audits per §5.

Lang — *Algebra* (Fast Track 3.01) — Audit + Gap Plan