Primary articles, transcribed and edited in 2000 by David R. Wilkins. Cited §§1–3 and §§10–11 for the units, associativity, distributivity and failure of commutativity. Original publication dates are listed in the editorial note.
Primary preface reproduced by the MacTutor archive. Its argument distinguishes arithmetical algebra from symbolical algebra and discusses the restrictions implicit in expressions such as .
The publisher's historical-edition description dates the original German book to 1770. The linked reproduction is an English edition. Cited for the subject's continuing association with general numerical operations and the solution of equations.
Original-edition title and metadata from New York Public Library. Used as evidence for the term universal arithmetic; the title alone does not define every contemporary meaning of algebra.
Tours original edition. The BnF catalogue establishes the title and date. Facsimile: gallica.bnf.fr/ark:/12148/bpt6k108865t . Folio 5r contrasts logistica numerosa and logistica speciosa; a scholarly reproduction and translation appears at www.uv.es/puigl/historias67.pdf . The latter is used for the specific terminology; the modern biographical discussion supplies the notation and homogeneity context.
Bulletin of the Australian Mathematical Society 40, 293–302. Primary historical research establishing the 1845 concentration of Cauchy's substitution work and explaining the misleading 1844 bibliographical dating. Publisher full text: www.cambridge.org/core/services/aop-cambridge-core/content/view/6801D1A609AA974ACB44B90C81533507/S000497270000438Xa.pdf/on-the-date-of-cauchys-contributions-to-the-founding-of-the-theory-of-groups.pdf .
Archive exposition based on a 19 March 2001 lecture by Peter Michael Neumann. Its account compares concrete substitutions, abstract definitions and Heinrich Weber's finite cancellation axioms. The byline explains the relation between the lecture and the written notes.
Mathematische Annalen 20, 301–329. The opening finite group definition is reproduced in Oxford's exam and discussed in The abstract group concept by the MacTutor History of Mathematics Archive. The publisher page establishes the article's title, author, pages and issue date; the commentary does not assume access to its subscription facsimile.
Original Latin memoir on biquadratic residues. ETH-Bibliothek record and facsimile, DOI doi.org/10.3931/e-rara-61067 . Presented in 1831; the linked original printing is dated 1832.
German self-announcement in Göttingische gelehrte Anzeigen. The digital transcription reproduces the argument about terminology, Gaussian integers and the complex plane. Its concluding discussion is the source of the wording quoted in the exam; this is distinct from the Latin memoir printed in 1832.
Revue d’histoire des mathématiques 19, 43–77. Original historical research separating Alexandre-Théophile Vandermonde's coefficient notation, alternating expressions, difference-products and the history of the Vandermonde determinant name. Cited especially §2.1 and the bibliography.
Published in the 1772 second-part academy volume, printed 1776, pp. 516–532. The linked research article by Bernard Ycart reproduces and explains the primary coefficient notation in §2.1 and supplies the original-edition reference. It should not be mistaken for a direct link to the complete eighteenth-century memoir.
By Leonhard Euler; English translation by Jordan Bell. Presented to the St Petersburg Academy on 26 September 1732; the original Latin paper, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus", appeared in Commentarii academiae scientiarum Petropolitanae 6 (1738), pp. 103–107. The publication note on page one of the translation distinguishes presentation from publication. The linked translation is version three, dated 15 April 2008.
Catalogue record: arxiv.org/abs/math/0501118
Cited passages: pp. 1–2, Euler's account of Fermat's proposed prime pattern and the factor of ; p. 1 publication and translator note. The short modular check in Fermat number is presented as a modern verification, not a historical reconstruction of Euler's search.
By Carl Pomerance. Slides for the ICERM meeting "Seventy-five years of Mathematics of Computation", 1–3 November 2018.
Cited passages: slide 14 states Pépin's test and outlines its proof; slides 15–16 describe Fermat numbers and their role in integer factorization; slide 18 gives the Gauss-Wantzel theorem and distinguishes Gauss's construction result from Wantzel's necessity proof. Numerical records elsewhere in these 2018 slides are historical records, not claims about current records.
Cited passages: Keith Conrad, Bilinear Forms, Theorem 1.7 and equation (1.3): symmetric and alternating parts.
Collection of sources; individual chapter authors and dates are recorded below.
Publisher: Terence Tao, What's new (author website).
Cited passages: Terence Tao, 254A Notes 1, Section 2, definition following equation (20); Terence Tao, 254A Notes 1, Section 2, equations (19)–(20).
Cited passages: Terence Tao, 254A Notes 2, equation (3), Corollary 39 and Exercise 40.
Collection of sources; individual chapter authors and dates are recorded below.
Publisher: Mathematics LibreTexts (verified host; original book publisher not established).
Cited passages: Wissam Raji, Elementary Number Theory, Section 7.2, definitions of theta and psi; Wissam Raji, Elementary Number Theory, Section 7.2, finite theta decomposition and Exercise 1.
Cited passages: Milne, Group Theory, corollary 4.8, printed page 61; Milne, Group Theory, example 4.6; Milne, corollary 4.8.
Additional cited passages: printed p. 57, Definition 4.1 and Example 4.2(c); p. 64, Action on the left cosets: stabilizer and kernel formulas preceding Remark 4.22, and Remark 4.22(a). James S. Milne is the verified sole author of the entire notes. The year 2025 identifies version 4.01; the cover and version-history dates differ by one day, so no exact publication day is assigned.
Publisher: Waveland Press, Inc..
Cited passages: Beachy and Blair, Abstract Algebra, second-edition excerpts, definition 7.2.1 and proposition 7.2.2.
Cited passages: Title and date Feb10 2014; §4 p3 Eq15–37, §5 p4 Eq38–42; p3 screenshot inspected.
Cited passages: §1 pp1–3 statements, historical Girard/Newton account; §4 pp6–8 logarithmic derivative proof; primary authored PDF text inspected.
Publisher: Cambridge University Press.
Cited passages: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 6.5, pp. 156–157, Theorem 6.29; multiplicative notation on p. 153; Victor Shoup, A Computational Introduction to Number Theory and Algebra, second edition, Section 6.5, pp. 153 and 156–157, Theorem 6.29.
Cited passages: Keith Conrad, Euler's Theorem, p. 1, Theorem 1.2; pp. 2–3, permutation of the units and cancellation.
Cited passages: Keith Conrad, Orders of Units in Modular Arithmetic, pp. 7–8, Theorem 4.5 and Remark 4.9.
Cited passages: Wilson theorem proof and preceding self-inverse lemma; Wilson's Theorem and Fermat's Theorem, Wilson theorem statement.
Cited passages: Notes on primitive lambda-roots, Section 5, pp. 7–8, definition; Section 5, p. 8, Proposition 5.1(a–c).
Cited passages: Definition 1.3, printed page 2: the two action axioms; Theorem 3.16(b)-(c), printed pages 12-13: conjugate stabilizers, transport cosets and the orbit-stabilizer formula.
Additional cited passages: printed p. 3, Theorem 1.7 and the following kernel paragraph; pp. 7–8, Example 2.12; p. 9, Definition 2.17 and Example 2.20; p. 20, comparison of free and faithful actions. The standalone notes are authored by Keith Conrad; the university hosts the author’s website. No publication year or version is verified.
Cited passages: Definition 1.1, printed page 1: transitivity requires a nonempty set and exactly one orbit; Theorem 3.1, printed page 2: equivalent tests for transitivity.
Published in Math. Comp..
Cited passages: Donald E. Knuth, Johann Faulhaber and sums of powers, 1992 preprint, pp. 2 and 14: the 1631 power-sum formulas and the uniform Bernoulli-number formula.
Publisher: Georgia Institute of Technology (online host and colophon copyright holder).
The front matter credits Dan Margalit and Joseph Rabinoff and is dated 3 June 2019. This is the online version date; the colophon copyright is 2017: textbooks.math.gatech.edu/ila/colophon-1.html
Section of Interactive Linear Algebra by Dan Margalit and others, by Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted. The parent record's 2019 year dates the front matter; the colophon copyright is 2017.
Cited passages: §6.3.1, Orthogonal decomposition theorem and Closest vector and distance; §6.3.2, Properties of Orthogonal Projections.
Cited passages: Margalit and Rabinoff, Interactive Linear Algebra, §2.4.2, Key Observation and Remark: translate the homogeneous solution set by any particular solution.
Cited passages: Margalit and Rabinoff, Interactive Linear Algebra, §2.9, Rank Theorem: column count equals rank plus nullity.
Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.
Cited passages: §1.2.1, Augmented Matrices and Row Operations: preservation of solutions; §1.2.2, Echelon Forms: leading entries and reduced form; §1.2.3, definitions of pivot position and pivot column.
Section of Interactive Linear Algebra by Dan Margalit and others by its whole-book authors Dan Margalit and Joseph Rabinoff; no separate chapter byline or date is asserted.
Cited passages: §3.2.1, One-to-one matrix transformations theorem and Wide matrices do not have one-to-one transformations; §3.2.2, Onto matrix transformations theorem and Tall matrices do not have onto transformations.
Collection of sources; individual chapter authors and dates are recorded below.
Publisher: MIT OpenCourseWare.
Cited passages: Gilbert Strang, Lecture 6: Column space and nullspace, opening lecture description: the distinct roles of Ax=0 and Ax=b.
Cited passages: Andrew Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach, §1.1.4, pp. 7–8: divisor pairing and the hyperbola split; §1.1.4, p. 8, equation (1.1.13): the divisor summatory asymptotic.
Publisher: Springer Cham.
Bibliographic and copyright year 2024; electronic release 28 October 2023 and print release 20 November 2023. The linked author PDF cover is dated 30 October 2026.
Cited passages: Axler, Linear Algebra Done Right, fourth edition, §3F, definitions 3.109–3.112; Axler, Linear Algebra Done Right, fourth edition, §3F, definitions and results 3.112–3.116; Axler, Linear Algebra Done Right, fourth edition, §8D, definition 8.47 and result 8.56; Axler, Linear Algebra Done Right, fourth edition, §8D, result 8.49; Axler, Linear Algebra Done Right, fourth edition, §8D, results 8.50–8.52; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3B, theorem 3.21; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3B: fundamental theorem of linear maps; Sheldon Axler, Linear Algebra Done Right, fourth edition, §3E: quotient spaces; §9D, 9.71–9.74, printed pp. 372–373: tensor products, bilinearity, and bases; §9D, Theorem 9.79(a), printed p. 375: unique linear factorization; §9D, Theorems 9.72 and 9.74, printed pp. 372–373: dimension and tensor-product basis.
Section 3A, definition 3.1, printed p. 52: additivity and homogeneity define a linear map. The fourth-edition bibliographic and copyright year remains 2024.
Publisher: Keith Conrad (author-hosted notes).
Cited passages: §3, printed p. 9: elementary, simple, decomposable, and pure tensors; Example 2.1, printed pp. 4–5: outer products do not fill the matrix space.
Publisher: Ben Lynn (hosted on Stanford Applied Cryptography site).
Cited passages: Ben Lynn, Quadratic Reciprocity, first theorem and proof of the Law of Quadratic Reciprocity: floor-sum parity and the separating-line count; Ben Lynn, Quadratic Reciprocity, “The Jacobi Symbol”: product definition and reciprocity for computing symbols without factoring.
Cited passages: Abstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley by Janet Heine Barnett, Section 3.1, printed page 50: reproduced opening paragraph of Arthur Cayley's On the theory of groups, as depending on the symbolic equation theta^n = 1; reference 5, printed page 91: original publication in 1854.
Cited passages: Keith Conrad, Fermat's Test, p. 2, Definition 2.4 and footnote 3 on the two witness conventions; Keith Conrad, Fermat's Test, pp. 5–6, Definition 4.1 and the list beginning with 561.
Cited passages: Theorem 1.1, pp. 1–2: even-number equivalence and proof; §1, Theorem 1.1 and Example 1.2, pp. 1–2.
Authored notes dated March 2011. Their running header identifies Master Course Diophantine Equations, Spring 2011. The supplied UConn mirror is hosted within a different Fall 2011 course by Keith Conrad; hosting and teaching that course do not confer authorship of these notes. No publisher or original publication date is inferred.
Cited passages: §3, Theorem 3.4 and digit/congruence discussion, printed pp. 7–9: unique expansion and agreement of low-order digits; §3, printed pp. 6–7: definition of Zp, rational denominators and units.
Standalone authored expository note. The inspected copy is undated; no publication date is inferred from the search engine or hosting path. It explicitly writes p-adic expansions with low-order terms first.
Cited passages: §1, printed p. 1: p-adic rational expansions and explicit low-order-first convention; §3, Theorem 3.1, printed pp. 3–5: eventual periodicity.
Collection of sources; individual chapter authors and dates are recorded below.
Lecture notes explicitly credited to Andrew V. Sutherland and dated 8 November 2021. The author footer credits this individual lecture; MIT is the institutional host and Fall 2021 identifies the course offering.
Cited passages: Lecture 17, §17.1, Lemma 17.9 and proof, printed p. 3: differential-equation proof for the unit Gaussian; scaled here to a>0; Lecture 17, §17.1, Theorem 17.8 and proof, printed p. 3: periodization and Fourier coefficients.
Collection of sources; individual chapter authors and dates are recorded below.
Lecture 1 notes credit Daniel Dadush as scribe, with Daniel Dadush and Léo Ducas listed as lecturers. Mastermath Spring 2018 identifies the course offering; no publication year or whole-course authorship is inferred.
Cited passages: §7, Definition 11 and Lemma 12, printed p. 9: determinant and integer-inverse characterization; §7, Lemma 13, printed p. 9: equivalent lattice bases.
Authored expository notes by Keith Conrad. No document date is established.
Cited passages: §§1–2, Theorem 2.5, printed pp. 1–4: rigid motions, rotations, reflections and dihedral order conventions.
Author-hosted expository notes, 47 pages. No date appears on the title page; no publication date is established.
Cited passages: §3, Definition 3.1 and Theorem 3.3; §4, Theorem 4.2; §10, exterior algebra.
By Keith Conrad. These standalone notes are hosted on the author's University of Connecticut website. No document date is established.
Cited passages: printed p. 1, title/byline and §1 opening paragraph: nonzero convention and equal counts of squares and nonsquares; printed p. 2, §2 opening notation for the Legendre symbol.
By Ben Lynn. Stanford hosts the author's PBC Notes; it is not an institutional author. The inspected page is undated.
Cited passages: “Quadratic Residues”, opening definition and Euler criterion discussion; “The Legendre Symbol”, three values, odd-prime restriction, multiplication and square factors. The opening square-residue convention includes zero.
By Keith Conrad. Standalone expository notes hosted on the author's University of Connecticut website. The inspected title page is undated; no publication year is established.
Cited passages: printed p. 1, Definition 1; printed p. 2, Theorem 2 and its proof; printed p. 3, Corollary 4 and its proof.
By Václav Šimerka. Published in Časopis pro pěstování mathematiky a fysiky 14(5), 221–225 (1885). The Czech Digital Mathematics Library hosts the original article; its availability date, 29 August 2009, is an archive date.
Cited passage: printed p. 224, §9, numerical list and factorizations.
By James S. Milne, credited as J.S. Milne on the cover and James S. Milne in the author-supplied BibTeX. Version 5.10, September 2022; this is a version date. The version history separately dates the first web version to 21 August 1996.
Cited passages: Chapter 1, Review of polynomial rings, printed p. 10, §1.6, unique polynomial division with zero remainder or strict polynomial degree bound; §1.7, division by a linear factor and evaluation. No separate chapter byline is supplied.
By Romyar Sharifi, credited on the introduction and chapter pages. UCLA hosts these lecture notes; no publication or revision date is established.
Chapter of Abstract Algebra by Romyar Sharifi, with the same Romyar Sharifi byline; no distinct contributed chapter author is established.
Cited passages: §3.9, Theorem 3.9.1 and proof, polynomial division when the leading coefficient is a unit; Proposition 3.9.7 and proof, the factor theorem. No publication date is inferred from the URL.
By Lynn Marecek and Andrea Honeycutt Mathis, whole-book authors. OpenStax publishes and hosts the book. Published 6 May 2020; the later website footer copyright year is not the publication date.
Section of Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis. No separate chapter author or section publication date is credited. The parent edition was published 6 May 2020.
Cited passages: §9.2, Binomial Squares Pattern; Solve Quadratic Equations of the Form ax²+bx+c=0; Citation information.
By David Anthony Santos, printed as David A. Santos on the title page and David Anthony Santos on the copyright page. The title page dates this revision 31 October 2007; the copyright year is separately 2007. The University of Texas Rio Grande Valley faculty website hosts the archived author text.
Cited passages: Title page and copyright page (PDF pp. 1–2); §8.3, Theorem 245 and proof, printed pp. 89–90 (PDF pp. 96–97); §8.3, Theorem 246 and proof, printed p. 90 (PDF p. 97).
By Keith Conrad. Standalone expository notes hosted on the author’s University of Connecticut website. No publication or revision date is established.
By Arthur Cayley. Published in Philosophical Transactions of the Royal Society of London 148, 17–37, 1858. Received 10 December 1857 and read 14 January 1858; these dates are distinct from journal publication. Wikimedia Commons hosts the archival scan.
Collected papers by James Joseph Sylvester, edited by Henry Frederick Baker, credited as H. F. Baker in the prefatory note dated April 1904. Cambridge University Press published Volume I in 1904. The collection date is distinct from the original dates of its individual papers. Internet Archive hosts the inspected digitized volume and OCR text.
Paper 9 by James Joseph Sylvester, reprinted on pp. 54–57 in The Collected Mathematical Papers of James Joseph Sylvester, Volume I. The reprint identifies the original publication as Philosophical Magazine XVI (1840), 132–135. Henry Frederick Baker edited the collection; he is not credited as author of this paper.
Cited passages: paper 9, printed pp. 54–55, rules 1–4, shifted coefficient rows and the permutation sign sum. The historical word derivatives in the title is not used here to mean differentiation in calculus. The title page and prefatory note establish the separate 1904 collection date.
By Yue Zhao. Preprint arXiv:1807.10125 (2018). Online text: arxiv.org/abs/1807.10125
Locator: §1, equation (1.4), printed p. 1.
Consulted locators: Chapter I, definition and basic properties; sections on the dual abelian variety, rigidity and the theorem of the cube.
Published in BIT, volume 20, pp. 176–184. Sections 3–4 develop the cycle-finding and factorization algorithms.
Cited passages: section I, pp. 565–567, equations (3), (6) and (8): bordered minors, exact divisibility and one-step integer-preserving update.
Official Macaulay2 documentation tutorial: www.macaulay2.com/doc/Macaulay2-1.23/share/doc/Macaulay2/Macaulay2Doc/html/___Tutorial_co_sp__Elementary_spuses_spof_sp__Groebner_spbases.html . The projective-closure discussion explains why homogenizing a chosen generating set can leave components at infinity and how saturation removes them.
Author-hosted notes, version 3.00 (1 April 2011): jmilne.org/math/CourseNotes/ALA.pdf . Section 14 treats characters, groups of multiplicative type and tori; Definition 14.24 defines a torus through splitting over a separable closure.
Original article, Journal of Algebraic Geometry4 (1995),17–50; author-hosted version: arxiv.org/pdf/alg-geom/9210008 . The inspected arXivv3 PDF includes the1993 revision and states the ray-span assumption in section1, the polynomial class grading and irrelevant ideal in section1, and categorical/geometric quotient theorem in section2.
Cited group page: brauer.maths.qmul.ac.uk/Atlas/lin/L27/L27.html, orders, automorphisms and maximal subgroups of L2(7) and its outer extension.
Cited passage: slide 7, strict inductive limits of closed topological subspace sequences of Hausdorff locally convex spaces. The slide title page identifies the presenter and date; this record cites the slides rather than attributing the joint research paper to one author.
Authorship: the document is linked as the bilinear-forms supplement from sites.math.rutgers.edu/~weibel/.
Cited passages: printed p. 2, classification of symmetric bilinear forms over an algebraically closed field; printed pp. 2–3, real diagonalization and Sylvester's law of inertia. The Witt-group examples in the wiki are deductions from these classifications, with the cancellation argument given explicitly.
Extended notes for MIT 18.781, Introduction to Number Theory, April 14, 2016. The current text contains sections on the floor function, greatest common divisors, arithmetic functions, Dirichlet convolution and Möbius inversion.
Published in CRM Proceedings and Lecture Notes, volume 46, pp. 205–211.
Published in the Journal of the American Mathematical Society 20(2), pp. 357–384. The article appeared electronically on May 26, 2006.
ChapterI, §8, printed p.102, defines the relative Weil group as a quotient by the closure of a commutator subgroup, identifies its extension class and describes its topology; Example8.1(a) specifies the local absolute Weil-group topology.
Mathematisch Centrum report ZW 1963-011, August 1963. Original institutional scan: ir.cwi.nl/pub/7151/7151D.pdf
Section 1, printed pages 3–5, displays the three unimodular generating matrices and the unique-generation theorem; the proof continues on page 7 after the inserted diagram/table. Barning puts the even leg first, with root (4,3,5).
The factorial-normalized polynomial, its integer endpoint derivatives and the positive integral tending to zero provide the complete elementary proof.
Proposition 1.1 and Remark 3.5 give full symmetric sectional monodromy in characteristic zero; Lemma 2.2 proves double transitivity. DOI of the journal publication: doi.org/10.1112/jlms.12375
Cited passages: Section 1.9, Definitions 1.9.1–1.9.2, coalgebras and comodules.
§1, Theorem 1 gives the factorization formula for the arithmetic derivative and verifies its defining product rule.
Theorem 2.10 gives the approximate-group structure theorem. Corollary 11.9 and its proof, pages 68–69 in the inspected author version, obtain a bounded-index finite-by-nilpotent structure from one-scale relative polynomial growth. The proof uses a smaller scale with bounded growth; fixed scale constants may be adjusted by increasing the threshold.
Published in Proceedings of the London Mathematical Society 107(3), pages 680-712.
Cited passages: introduction, exceptional Chevalley-group automorphisms and characteristic restrictions; section 3.1, automorphism-group constructions; sections 4-5, orders, doubly transitive actions and simplicity.
Cited passages: chapter 50, section 50.1-13, MathieuGroup and its explicit degree-eleven generators; sections 50.1-14 and 50.1-15, SuzukiGroup and ReeGroup parameter conventions.
Cited passage: Mathieu groups and the coordinate-permutation automorphism group of the Golay code.
Summer tutorial notes. Section 2.2, Definition 1 and Theorem 2.3 state primitive positive definite reduction and uniqueness; the following argument bounds the reduced coefficients. The primitive-form hypotheses must be retained when counting a class number of a negative discriminant.
Theorem 15, equations (3.41)–(3.42), printed pp. 71–72, give the Rosser–Schoenfeld totient bound. The coefficient 2.5 has an exception at 223092870; 2.50637 covers that case as well. Section 9 contains the proof.
By Patrice Philippon, Michel Waldschmidt. Chapter 18 of New Advances in Transcendence Theory, pages 280–312, Cambridge University Press.
Theorem 1.1, printed pages 280–281, treats algebraic coefficients including the constant term. Theorems 2.1–2.2, printed pages 282–283, distinguish general algebraic coefficients from the homogeneous rational-integer case and give the refined coefficient parameter.
Corollary 19 states the fundamental lemma of sieve theory with a summed remainder; equations (26) and Corollary 21 apply it to simultaneous almost prime values. The opening definitions and the quadratic-weight construction describe sieve theory and the Selberg sieve.
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