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Commentary on an unpublished lecture by G. N. Watson on solving the quintic

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References

  1. Bruce C. Berndt,Ramanujan’s Notebooks, Springer-Verlag, New York, Part I (1985), Part II (1989), Part III (1991), Part IV (1994), Part V(1998).

    Book  MATH  Google Scholar 

  2. Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ramanujan’s class invariants and cubic continued fraction,Acta Arithmetica 73 (1995), 67–85.

    MATH  MathSciNet  Google Scholar 

  3. Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations,Transactions of the American Mathematical Society 349 (1997), 2125–2173.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruce C. Berndt, Youn-Seo Choi, and Soon-Yi Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, inContinued Fractions: From Analytic Number Theory to Constructive Approximation, B. C. Berndt and F. Gesztesy, eds., Contemp. Math. No. 236, American Mathematical Society, Providence, Rl, 1999, pp. 15–56.

    Chapter  Google Scholar 

  5. William S. Burnside and Arthur W. Panton,The Theory of Equations, 2 vols., Dover, New York, 1960.

    Google Scholar 

  6. Arthur Cayley, On a new auxiliary equation in the theory of equations of the fifth order,Philosophical Transactions of the Royal Society of London CLI (1861), 263–276. [15, Vol. IV, Paper 268, pp. 309-324.]

    Article  Google Scholar 

  7. Arthur Cayley, Note on Mr. Jerrard’s researches on the equation of the fifth order,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXI (1861), 210–214. [15, Vol. V, Paper 310, pp. 50-54.]

    Google Scholar 

  8. Arthur Cayley, On a theorem of Abel’s relating to equations of the fifth order,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXI (1861),257–2633. [15, Vol. V, Paper 311, pp. 55-61.]

    Google Scholar 

  9. Arthur Cayley, Note on the solution of an equation of the fifth or- der,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXIII (1862), pp. 195, 196. [15, Vol. V, Paper 316, p. 77.]

    Google Scholar 

  10. Arthur Cayley, Final remarks on Mr. Jerrard’s theory of equations of the fifth order,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXIV (1862), 290. [15, Vol. V, Paper 321, p. 89.]

    Google Scholar 

  11. Arthur Cayley, Note on the solvability of equations by means of radicals,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXXVI (1868), pp. 386, 387. [15, Vol. VII, Paper 421, pp. 13-14.]

    Google Scholar 

  12. Arthur Cayley, On a theorem of Abel’s relating to a quintic equation,Cambridge Philosophical Society Proceedings III (1880), 155–159. [15, Vol. XI, Paper 741, pp. 132-135.]

    Google Scholar 

  13. Arthur Cayley, A solvable case of the quintic equation,Quarterly Journal of Pure and Applied Mathematics XVIII (1882), 154–157. [15, Vol. XI, Paper 777, pp. 402-404.]

    Google Scholar 

  14. Arthur Cayley, On a soluble quintic equation,American Journal of Mathematics XIII (1891), 53–58. [15, Vol. XIII, Paper 914, pp. 88-92.]

    MathSciNet  Google Scholar 

  15. Arthur Cayley,The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, Vol. I (1889), Vol. II (1889), Vol. III (1890), Vol. IV (1891), Vol. V (1892), Vol. VI (1893), Vol. VII (1894), Vol. VIII (1895), Vol. IX (1896), Vol. X (1896), Vol. XI (1896), Vol. XII (1897), Vol. XIII (1897), Vol. XIV (1898).

  16. Heng Huat Chan, Ramanujan-Weber class invariantG nand Watson’s empirical process,Journal of the London Mathematical Society 57 (1998), 545–561.

    Article  Google Scholar 

  17. James Cockle, Researches in the higher algebra,Memoirs of the Literary and Philosophical Society of Manchester XV (1858), 131–142.

    Google Scholar 

  18. James Cockle, Sketch of a theory of transcendental roots,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XX (1860), 145–148.

    Google Scholar 

  19. James Cockle, On the resolution of quintics,Quarterly Journal of Pure and Applied Mathematics 4 (1861), 5–7.

    Google Scholar 

  20. James Cockle, Notes on the higher algebra,Quarterly Journal of Pure and Applied Mathematics 4 (1861), 49–57.

    Google Scholar 

  21. James Cockle, On transcendental and algebraic solution-supplementary paper,The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXIII (1862), 135–139.

    Google Scholar 

  22. Winifred A. Cooke, George Neville Watson,Mathematical Gazette 49 (1965), 256–258.

    Google Scholar 

  23. David A. Cox,Primes of the Form x 2 + ny2, Wiley, New York, 1989.

    Google Scholar 

  24. David S. Dummit, Solving solvable quintics,Mathematics of Computation 57 (1991), 387–401.

    Article  MATH  MathSciNet  Google Scholar 

  25. David S. Dummit and Richard M. Foote,Abstract Algebra, Prentice Hall, New Jersey, 1991.

    MATH  Google Scholar 

  26. W. L. Ferrar,Higher Algebra, Oxford University Press, Oxford, 1950.

    Google Scholar 

  27. Joseph A. Gallian,Contemporary Abstract Algebra, Fourth Edition, Houghton Mifflin Co., Boston MA, 1998.

    MATH  Google Scholar 

  28. J. C. Glashan, Notes on the quintic,American Journal of Mathematics 7 (1885), 178–179.

    Article  MATH  MathSciNet  Google Scholar 

  29. Robert Harley, On the method of symmetric products, and its application to the finite algebraic solution of equations,Memoirs of the Literary and Philosophical Society of Manchester XV (1859), 172–219.

    Google Scholar 

  30. Robert Harley, On the theory of quintics,Quarterly Journal of Pure and Applied Mathematics 3 (1859), 343–359.

    Google Scholar 

  31. Robert Harley, On the theory of the transcendental solution of algebraic equations,Quarterly Journal of Pure and Applied Mathematics 5(1862), 337–361.

    Google Scholar 

  32. R. Bruce King,Beyond the Quartic Equation, Birkhäuser, Boston, 1996.

    MATH  Google Scholar 

  33. Sigeru Kobayashi and Hiroshi Nakagawa, Resolution of solvable quintic equation,Mathematica Japonicae 37 (1992), 883–886.

    MATH  MathSciNet  Google Scholar 

  34. John Emory McClintock, On the resolution of equations of the fifth degree,American Journal of Mathematics 6 (1883-1884), 301- 315.

    Article  MATH  MathSciNet  Google Scholar 

  35. John Emory McClintock, Analysis of quintic equations,American Journal of Mathematics 8 (1885), 45–84.

    Article  MATH  MathSciNet  Google Scholar 

  36. John Emory McClintock, Further researches in the theory of quintic equations,American Journal of Mathematics 20 (1898), 157–192.

    Article  MATH  MathSciNet  Google Scholar 

  37. Srinivasa Ramanujan, Modular equations and approximations to π,Quarterly Journal of Mathematics 45 (1914), 350–372. [40: pp. 23-39.]

    Google Scholar 

  38. Srinivasa Ramanujan, Question 699,Journal of the Indian Mathematical Society 7 (1917), 199. [40: p. 331.]

    Google Scholar 

  39. Srinivasa Ramanujan,Notebooks, 2 vols., Tata Institute of Fundamental Research, Bombay, 1957.

    MATH  Google Scholar 

  40. Srinivasa Ramanujan,Collected Papers of Srinivasa Ramanujan, AMS Chelsea, Providence, Rl, 2000.

    MATH  Google Scholar 

  41. Robert A. Rankin, George Neville Watson,Journal of the London Mathematical Society 41 (1966), 551–565.

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Russell, On modular equations,Proceedings of the London Mathematical Society 21 (1889-1890), 351–395.

    Google Scholar 

  43. Blair K. Spearman and Kenneth S. Williams, Characterization of solvable quintics x5 + ax + b,American Mathematical Monthly 101 (1994), 986–992.

    Article  MATH  MathSciNet  Google Scholar 

  44. Blair K. Spearman and Kenneth S. Williams, DeMoivre’s quintic and a theorem of Galois,Far East Journal of Mathematical Sciences 1 (1999), 137–143.

    MATH  MathSciNet  Google Scholar 

  45. Blair K. Spearman and Kenneth S. Williams, Dihedral quintic polynomials and a theorem of Galois,Indian Journal of Pure and Applied Mathematics 30 (1999), 839–845.

    MATH  MathSciNet  Google Scholar 

  46. Blair K. Spearman and Kenneth S. Williams, Conditions for the insolvability of the quintic equation x5 + ax + b = 0,Far East Journal of Mathematical Sciences 3 (2001), 209–225.

    MATH  MathSciNet  Google Scholar 

  47. Blair K. Spearman and Kenneth S. Williams, Note on a paper of Kobayashi and Nakagawa,Scientiae Mathematicae Japonicae 53 (2001), 323–334.

    MATH  MathSciNet  Google Scholar 

  48. K. L. Wardle, George Neville Watson,Mathematical Gazette 49 (1965), 253–256.

    MATH  Google Scholar 

  49. George N. Watson, Solution to Question 699,Journal of the Indian Mathematical Society 18 (1929-1930), 273–275.

    Google Scholar 

  50. George N. Watson, Theorems stated by Ramanujan (XIV): a singular modulus,Journal of the London Mathematical Society 6 (1931), 126–132.

    Article  Google Scholar 

  51. George N. Watson, Some singular moduli (I),Quarterly Journal of Mathematics 3 (1932), 81–98.

    Article  Google Scholar 

  52. George N. Watson, Some singular moduli (II),Quarterly Journal of Mathematics 3 (1932), 189–212.

    Article  Google Scholar 

  53. George N. Watson, Singular moduli (3),Proceedings of the London Mathematical Society 40 (1936), 83–142.

    Article  Google Scholar 

  54. George N. Watson, Singular moduli (4),Acta Arithmetica 1 (1936), 284–323.

    Google Scholar 

  55. George N. Watson, Singular moduli (5),Proceedings of the London Mathematical Society 42 (1937), 377–397.

    Article  Google Scholar 

  56. George N. Watson, Singular moduli (6),Proceedings of the London Mathematical Society 42 (1937), 398–409.

    Article  MATH  Google Scholar 

  57. Heinrich Weber,Lehrbuch der Algebra, 3 vols., Chelsea, New York, 1961.

    Google Scholar 

  58. George P. Young, Resolution of solvable equations of the fifth degree,American Journal of Mathematics 6 (1883-1884), 103–114.

    Article  Google Scholar 

  59. George P. Young, Solution of solvable irreducible quintic equations, without the aid of a resolvent sextic,American Journal of Mathematics 7 (1885), 170–177.

    Article  MATH  MathSciNet  Google Scholar 

  60. George P. Young, Solvable quintic equations with commensurable coefficients,American Journal of Mathematics 10 (1888), 99–130.

    Article  MathSciNet  Google Scholar 

  61. Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula and modular equations (II), inAnalytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Vol. 2, B. C. Berndt, H. G. Diamond and A. J. Hildebrand, eds., Birkhäuser, Boston, 1996, pp. 817–838.

    Google Scholar 

  62. Liang-Cheng Zhang, Ramanujan’s class invariants, Kronecker’s limit formula and modular equations (III),Acta Arithmetica 82 (1997), 379–392.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, Illinois, USA

    Bruce C. Berndt

  2. Department of Mathematics and Statistics, Okanagan University College Kelowna, V1V 1V7, British Columbia, Canada

    Blair K. Spearman

  3. School of Mathematics and Statistics, Carleton University Ottawa, K1S 5B6, Ontario, Canada

    Kenneth S. Williams

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  1. Bruce C. Berndt
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  2. Blair K. Spearman
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  3. Kenneth S. Williams
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Correspondence to Bruce C. Berndt.

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Berndt, B.C., Spearman, B.K. & Williams, K.S. Commentary on an unpublished lecture by G. N. Watson on solving the quintic. The Mathematical Intelligencer 24, 15–33 (2002). https://doi.org/10.1007/BF03025320

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