Bibliography

Below is a list of works relevant to the seminar. To avoid flagrant copyright violations, some of the works are in a password protected directory. This bibliography is currently work in progress, and will be filled out over the course of the semester.

General references

Primary sources

• Euler, Theorems on residues obtained by the division of powers, E262 on the Euler archive.
• Gauss, Disquisitiones Arithmeticae, 1801: GDZ. Translation by Arthur A. Clarke, Yale University Press, 1965: publisher. Text, through III, 66 together with 76 (Wilson's theorem): pdf.
• Dirichlet, Peter Gustav Lejeune, Vorlesungen über Zahlentheorie (edited by Richard Dedekind), 1837. Translated with introductory notes by John Stillwell as Lectures on Number Theory, AMS, 1999. Stillwell's introduction is available online.
• Weber, Heinrich, Lehrbuch der Algebra, Vieweg, Braunschweig, 1895.

Secondary sources

• Corry, Leo, Modern Algebra and the Rise of Mathematical Structures, Birkhäuser, 1996. Chapters 1, 2, 3.1-3.2, 4.1-4.2, and 5. Excerpts: Chapter 1: pdf, Chapter 2: pdf, Chapter 3: pdf.
• Edwards, Harold, Fermat's Last Theorem: a genetic introduction to algebraic number theory, Springer Verlag, 1977.
• Goldman, Jay, The Queen of Mathematics: a historically motivated guide to number theory. A K Peters, 1998.
• Goldstein et al., eds., The Shaping of Arithmetic after C. F. Gauss' Disquisitiones Arithmeticae, Springer, 2007.
• Knoebel et al, Mathematical Masterpieces, Springer, 2007. Chapter 4, Patterns in the Prime Numbers: The Quadratic Reciprocity Law: pdf.
• Stahl, Saul, Introductory Modern Algebra: A Historical Approach, John Wiley & Sons, New York, 1996.
• Tignol, Jean-Pierre, Galois' Theory of Algebraic Equations, World Scientific, 2001.
• Weil, André , Number theory: an approach through history, from Hamumurapi to Legendre. Birkhaüser, 1984.

Mathematical background

• Birkhoff, Garret, and Saunders Mac Lane, A Survey of Modern Algebra, third edition, Macmillan, New York, 1965.
• Ireland, Kenneth and Michael Rosen, A Classical Introduction to Modern Number Theory, second edition, Springer, 1990
• Manin, Yu. I. and Alexei A. Panchishkin, Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories, Springer, 2005.
• Milne, J.S., Fields and Galois theory: html
• Pollard, Harry and Harold Diamond, The Theory of Algebraic Numbers, 1998 Dover reprinting.
• Stewart, Ian and David Tall, Algebraic Number Theory and Fermat's Last Theorem, third edition, AK Peters, 2002.
• Swinnerton-Dyer, Peter, A Brief Guide to Algebraic Number Theory, Cambridge, 2001.

Useful sites

• Euler archive.
• Gallica digital library.
• Göttinger Digitalisierungzentrum.
• Mactutor history of mathematics archive.
• Cornell historical math monographs.

Quadratic reciprocity

Primary sources

• Gauss, DA, Section IV, 94-120: pdf.
• Gauss, third proof of quadratic reciprocity: pdf. English translation, from Struik's A Source Book in Mathematics: pdf
• Gauss, all six published proofs translated into German by Netto: pdf
• Cayley's translation of Eisenstein's proof: pdf
• Eisenstein's second and fourth proofs (from Vol II of his Werke): pdf
• Dedekind, Supplement X, ...
• Dedekind, 1877 monograph, section 27.

Secondary sources

• Brown, Ezra, "The First Proof of the Reciprocity Law, Revisited." American Mathematical Monthly, vol 88, pp. 257-64. JSTOR.
• Cox, David, "Quadratic Reciprocity: Its Conjecture and Application," American Mathematical Monthly, vol. 95, pp. 442-448. JSTOR, pdf.
• Davenport, H, The Higher Arithmetic, excerpt on quadratic reciprocity: pdf.
• Edwards, Harold, "Euler and Quadratic Reciprocity," Mathematics Magazine, vol. 56, pp. 285-291. JSTOR, pdf.
• Everest and Ward, An introduction to number theory, Springer, 2005, excerpt on quadratic reciprocity: pdf.
• R. Lautenbacher and D. Pengelley, "Eisenstein's Misunderstood Geometric Proof of the Quadratic reciprocity Theorem." The College Mathematics Journal (25) 1994, pp. 29--34. JSTOR, pdf.
• Lemmermeyer, Franz, Reciprocity Laws from Euler to Eisenstein, Springer, 2000.
• Serre, J.P., excerpt from A course in arithemetic: pdf.
• Tate, John, "Problem 9: The general reciprocity law". In F. Browder,ed., Mathematical developments arising from HILBERT PROBLEMS. AMS, 1976, Part 2, p.311-322. pdf.
• Webster, Ben, Math overflow posting.
• Wyman, B. F., "What is a reciprocity law?," American Mathematical Monthly, 79:571-586, 1972: JSTOR.
• Zuser, R., [233] Proofs of the Quadratic Reciprocity Law.

Quadratic forms and the theory of algebraic integers

Primary sources

• Euler, Theorems on divisors of numbers, E134 on the Euler archive.
• Gauss, DA, Section V. Excerpt: pdf.
• Dirichlet, P. G. Lejeune, De formarum binariarum secundi gradus compositione, 1851.
• Dedekind's preface to the second edition of Dirichlet's Lectures on Number Theory. Draft translation by Jeremy Avigad: pdf.
• Dedekind, Richard, Sur la théorie des nombres entiers algébrique, translated with introductory notes by John Stillwell as Lectures on the Algebraic Integers, Cambridge University, 1996. Introduction: pdf, part 1: pdf, part 2: pdf, part 3: pdf.
• Dirichlet's Lectures on Number Theory, Chapter 4, On quadratic forms. Stillwell translation: pdf.
• Dedekind, Richard, treatment of composition of binary quadratic forms in Supplement X to the second edition of Dirichlet's Vorlesungen. Draft translation by Jeremy Avigad: pdf.
• Dedekind, Richard, treatment of ideal theory in the same supplement. Translation by Jeremy Avigad, with an introduction: pdf.

Secondary sources

• Avigad, Jeremy, "Mathematical method and proof," Synthese 153:105-159, 2006 (specifically sections 2.2 and 2.3): doi, pdf.
• Avigad, Jeremy, "Methodology and metaphysics in the development of Dedekind's theory of ideals," in José Ferreirós and Jeremy Gray, editors, The Architecture of Modern Mathematics, Oxford University Press: pdf.
• Buell, Duncan A., Binary quadratic forms, Springer, 1989.
• Cohn, Harvey, Advanced Number Theory, Dover reprint, 1980. Excerpt on the relationship between ideals and quadratic forms: pdf.
• Edwards, Fermat's Last Theorem, Chapters 7 and 8. Excerpt on composition of forms: pdf.
• Edwards, Harold, "Dedekind's invention of ideals," Bulletin of the London Mathematics Society, 15:8-17, 1983. Also in E. R. Phillips, editor, Studies in the History of Mathematics, MAA, 1988.
• Edwards, Harold, "Mathematical Ideas, Ideals, and Ideology," The Mathematical Intelligencer, 14:6-19, 1992: journal, >pdf.
• Goldman, Chapter 5, Lagrange: pdf.
• Goldman, Chapter 12, Binary Quadratic Forms I: pdf.
• Excerpts from Weil, on Brahmagupta's identity and Legendre's composition of quadratic forms: pdf.

Cyclotomy and Galois theory

Primary sources

• Gauss, DA, Section VII.
• Abel, Niels Henrik, "Mémoire sur les équations algébriques, où l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré," 1824. In his Ouvres, 1:28-33. Pesic translation: pdf.
• Abel, Niels Henrik, "Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré," 1826. In his Ouvres, 1:66-86. German version published in Crelle's Journal, volume 1, 1826.
• Abel, Niels Henrik, summary of the preceding article, published in Bulletin des sciences math., astr., phys., et chim. 6:347, 1826. In his Ouvres, 1:87-94. Manders translation: pdf.
• Galois, Évariste, "Mémoire sur les conditions de resolubilité des equations par radicaus,"1831-1832. Translation by Harold Edwards. pdf.
• Dedekind, Richard, 1857 lecture notes on Galois theory, translation by Edward Dean: pdf.
• Dedekind, Richard, Suplement XI to the fourth edition Dirichlet's Vorlesungen über Zahlentheorie, Vieweg, Braunschweig, 1894. Reprinted in Dedekind's Werke, volume 3, 1-222. Excerpt: pdf. Translation by Edward Dean, with an introduction: pdf.
• Wantzel, M.L. "Recherches sur les moyens de reconnaitre si un Problème de Géométrie peut se résoudre avec la règle et le compas". Journal de Mathématiques Pures et Appliquées 1: 366-372, 1837: pdf.
• Weber ...

Secondary sources

• Edwards, Harold M., Galois Theory, Springer-Verlag, New York, 1984. Excerpts: pdf, pdf, pdf.
• Edwards, excerpt on cyclotomic integers from Fermat's last theorem: pdf.
• Kiernan, B. M., "The development of Galois theory from Lagrange to Artin," Archive for History of Exact Science 8, 1971, 40-154. pdf.
• Pesic, Peter, Abel's Proof: an essay on the sources and meaning of mathematical unsolvability, MIT Press, Cambridge, MA, 2003.
• Tignol's book, Chapter 11, Vandermonde: pdf.
• Tignol's book, Chapter 12, Gauss on cyclotomy: pdf.
• Tignol's book, Chapter 14, Galois: pdf.

Miscellaneous

• Sandborg, David, Explanation in Mathematical Practice, PhD thesis, University of Pittsburgh, 1997. Chapter 4: (and title page) pdf, Chapter 5: pdf, Chapter 6: pdf, references: pdf.
• Niven, Ivan and H. S. Zuckerman, "Lattice points and polygonal area," AMS Monthly, 74:1195-1200, 1967. JSTOR, pdf.
• Kleiner, Israel, "From Numbers to Rings: the early history of ring theory," Elemente der Mathematik, 53:12-35, 1998.