Skip to main content
SpringerLink
Log in

Galois Theories: A Survey

  • Chapter
  • First Online:
Book cover Advances in Mathematics and Applications

  • 684 Accesses

Abstract

We present an overview of the various Galois theories that appeared in the literature since Évariste Galois until to the present day, accompanied with a bit of the inherent history.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. F. Abadie, Enveloping actions and Takay duality for partial actions, J. Funct. Analysis 197 (2003), 14–67.

    Article  MathSciNet  Google Scholar 

  2. M. Artin, Grothendieck Topologies, Notes on a Seminar, Harvard University, 1962.

    MATH  Google Scholar 

  3. E. Artin, Galois Theory, Notre Dame Mathematical Lectures 2, First edition in 1942, Dover Books on Mathematics, Sixth edition in 1971.

    Google Scholar 

  4. G. Azumaya, Galois theory for uni-serial rings, J. Math. Soc. Japan 1 (1949), 130–137.

    Article  MathSciNet  Google Scholar 

  5. M. M. S. Alves, E. Batista, Partial Hopf actions, partial invariants and a Morita context, Algebra Disc. Math. no. 3, (2009), 1–19.

    MathSciNet  MATH  Google Scholar 

  6. M. M. S. Alves, E. Batista, Enveloping Actions for Partial Hopf Actions, Comm. Algebra 38 (2010),2872–2902.

    Article  MathSciNet  Google Scholar 

  7. M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. AMS. 97 (1960), 367–409.

    Article  MathSciNet  Google Scholar 

  8. J. Ávila Guzmán, J. Lazzarin, A Morita context related to finite groups acting partially on a ring, Algebra Disc. Math. 3. (2009), 49–61.

    Google Scholar 

  9. D. Azevedo, E. Campos, G. Fonseca and G. Martini, Partial (Co)Actions of Multiplier Hopf Algebras: Morita and Galois Theories, arXiv: 1709.08051, (2017). 32 pages.

    Google Scholar 

  10. D. Bagio, W. Cortes, M. Ferrero and A. Paques, Actions of inverse semigroups on algebras, Comm. Algebra 35 (2007), 3865–3874.

    Article  MathSciNet  Google Scholar 

  11. D. Bagio, D. Flores and A. Paques, Partial actions of ordered groupoids on rings, J. Algebra Appl. 9 (2010), (3), 501–517.

    Google Scholar 

  12. D. Bagio, J. Lazzarin and A. Paques, Crossed products by twisted partial actions: separability, semisimplicity and Frobenius properties, Comm. Algebra 38 (2010), 496–508.

    Article  MathSciNet  Google Scholar 

  13. D. Bagio and A. Paques, Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012), 3658–3678.

    Article  MathSciNet  Google Scholar 

  14. E. Batista, Partial actions: What they are and why we care, Bull. Belg. Math. Soc. -Simon Stevin, 24, (2017), (1), 35–71 (arXiv:1604.06393v1).

    Google Scholar 

  15. F. Borceux and G. Janelidze, Galois Theories, Cambridge Univ. Press, 2001.

    Book  Google Scholar 

  16. S. Caenepeel, Galois corings from the descent theory point of view, Fields Inst. Comm. 43 (2004), 163–186.

    MathSciNet  MATH  Google Scholar 

  17. S. Caenepeel and E. De Groot, Corings applied to partial Galois theory, Proceedings of the International Conference on Mathematics and Applications, ICMA 2004, Kuwait Univ. (2005), 117–134.

    Google Scholar 

  18. S. Caenepeel and E. De Groot, Galois theory for weak Hopf algebras, Rev. Roumaine Math. Pures Appl. 52 (2007), 51–76.

    MathSciNet  MATH  Google Scholar 

  19. S. Caenepeel and K. Janssen, Partial (co)actions of Hopf algebras and partial Hopf-Galois theory, Comm. Algebra 36 (2008), 2923–2946.

    Article  MathSciNet  Google Scholar 

  20. H. Cartan, Théorie de Galois pour les corps non-commutatifs, Ann. Scien. de l’École Norm. Sup. 65 (1948), 60–77

    Google Scholar 

  21. F. Castro, A. Paques, G. Quadros and A. Sant’Ana, Partial actions of weak Hopf algebras: Smash product, globalization and Morita theory, J. Pure Appl. Algebra, 219, (2015), 5511–5538.

    Article  MathSciNet  Google Scholar 

  22. F. Castro and G. Quadros, Galois Theory for Partial Comodule Coalgebras, Preprint.

    Google Scholar 

  23. S. U. Chase, D. K. Harrison and A. Rosenberg, Galois theory and Galois cohomology of commutative rings, Memoirs AMS 52 (1965), 1–19.

    MathSciNet  MATH  Google Scholar 

  24. S. U. Chase and M. E. Sweedler, Hopf Algebras and Galois Theory, LNM 97, Springer Verlag, Berlin, 1969.

    Book  Google Scholar 

  25. C. Cibils and A. Solotar, Galois coverings, Morita equivalence and smash extensions of categories over a field, Doc. Math., 11, (2006), 143–159.

    MathSciNet  MATH  Google Scholar 

  26. M. Cohen, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98 (1) (1982).

    Google Scholar 

  27. M. Cohen, D. Fischman and S. Montgomery, Hopf-Galois extensions, smash products and Morita equivalence, J. Algebra 133 (1990), 351–372.

    Article  MathSciNet  Google Scholar 

  28. W. Cortes and E. Marcos, Descriptions of partial actions, Preprint, arXiv:1708.01330, (2017).

    Google Scholar 

  29. W. Cortes and T. R. Tamusiunas, A characterisation for a groupoid Galois extension using partial isomorphisms, Bull. Aust. Math. Soc. (2017), 1–10.

    Google Scholar 

  30. F. DeMeyer, Galois Theory in Rings and Algebras, PhD thesis, Univ. of Oregon, 1965, 59 pp.

    Google Scholar 

  31. F. DeMeyer, Some notes on the general Galois theory of rings, Osaka J. Math. 2 (1965), 117–127.

    MathSciNet  MATH  Google Scholar 

  32. F. DeMeyer and E. Ingraham, Separable Algebras Over Commutative Rings, LNM 181, Springer Verlag, Berlin, 1971.

    Book  Google Scholar 

  33. J. Dieudonné, La théorie de Galois des anneaux simples et semi-simples, Comment. Math. Helv. 21 (1948), 154–184.

    Article  MathSciNet  Google Scholar 

  34. M. Dokuchaev, Partial actions: A survey. Contemporary Math. 537 (2011), 173–184.

    Article  MathSciNet  Google Scholar 

  35. M. Dokuchaev, Recent developments around partial actions, arXiv:1801.09105v1 (2018), 60 pages.

    Google Scholar 

  36. M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. AMS 357 (2005) 1931–1952.

    Article  MathSciNet  Google Scholar 

  37. M. Dokuchaev, M. Ferrero and A. Paques, Partial actions and Galois theory, J. Pure Appl. Algebra 208 (2007), 77–87.

    Article  MathSciNet  Google Scholar 

  38. Y. Doi and M. Takeuchi, Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, J. Algebra 121 (1989), 448–516.

    Article  MathSciNet  Google Scholar 

  39. A. W. Dress, One more shortcut to Galois theory, Adv. Math. 110 (1995), 129–140.

    Article  MathSciNet  Google Scholar 

  40. A. W. Dress, Basic non-commutative Galois Theory, Adv. Math. 110 (1995), 141–173.

    Article  MathSciNet  Google Scholar 

  41. R. Exel, Twisted partial actions: a classification of regular C ⋆-algebraic bundles, Proc. London Math. Soc. 74 (3) (1997) 417–443.

    Article  MathSciNet  Google Scholar 

  42. R. Exel, Partial actions of groups and actions of semigroups, Proc. AMS 126 (1998) 3481–3494.

    Article  MathSciNet  Google Scholar 

  43. R. Exel, Partial Dynamical Systems, Fell Bundles and Applications, Mathematical Surveys and Monographs, American Mathematical Society, vol. 224, 321 pages, 2017.

    Google Scholar 

  44. V. Ferreira, L. Murakami and A. Paques, A Hopf-Galois correspondence for free algebras, J. Algebra 276 (2004), 407–416.

    Article  MathSciNet  Google Scholar 

  45. M. Ferrero, Teoria de Galois para Anillos Graduados, PhD Thesis, Univ. de Buenos Aires, 1970.

    Google Scholar 

  46. M. Ferrero, On the Galois theory over non-comutative rings, Osaka J. Math. 7 (1970), 81–88.

    MathSciNet  MATH  Google Scholar 

  47. M. Ferrero and A. Paques, Galois theory of commutative rings revisited, Beiträge zur Algebra und Geometrie 38 (1997), 299–410.

    MathSciNet  MATH  Google Scholar 

  48. G. L. Fonseca, Galois, Dedekind e Grothendieck, MSc dissertation, UFRGS, Brazil, 2015.

    Google Scholar 

  49. D. Freitas and A. Paques, On partial Galois Azumaya extensions, Algebra Disc. Math., 11, (2011), 64–77.

    MathSciNet  MATH  Google Scholar 

  50. A. Grothendieck, Revêtements étales et groupe fondamental, SGA 1, exposé V, LNM 224, Springer Verlag, (1971).

    Google Scholar 

  51. G. Hochschild, Double vetor spaces over division rings, Amer. J. Math. 71 (1949), 443–460.

    Article  MathSciNet  Google Scholar 

  52. N. Jacobson, The fundamental theorem of Galois theory for quasi-fields, Annals Math. 41 (1940), 1–7.

    Article  MathSciNet  Google Scholar 

  53. X.-L. Jiang and G. Szeto, Galois extensions induced by a central idempotent in a partial Galois extension, Intern. J. Algebra, 8, (11), (2014), 505–510.

    Article  Google Scholar 

  54. X.-L. Jiang and G. Szeto,The group idempotents in a partial Galois extension, Gulf J. Math. 3 (2015), 42–47.

    Google Scholar 

  55. T. Kanzaki, On commutator rings and Galois theory of separable algebras, Osaka J. Math. 1 (1964), 103–115.

    MathSciNet  MATH  Google Scholar 

  56. T. Kanzaki, On Galois extension of rings, Nagoya Math. J. 27 (1966), 43–49.

    Article  MathSciNet  Google Scholar 

  57. V. Kharchenko, Galois theory of semiprime rings, Algebra i Logica 16 (1977), 313–363 (English translation 1978, 208–258).

    Google Scholar 

  58. E. R. Kolchin, Picard-Vessiot theory of partial differential fields, Proc. AMS 3 (1952), 596–603.

    Article  MathSciNet  Google Scholar 

  59. H. F. Kreimer, A Galois theory of non-commutative rings, Trans. AMS 127 (1967), 29–41.

    Article  Google Scholar 

  60. H.F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), 675–692.

    Article  MathSciNet  Google Scholar 

  61. W. Krull, Galoissche Theorie der unendlichen algebraschen Erweiterungen, Math. Ann. 100 (1928), 687–698.

    Article  MathSciNet  Google Scholar 

  62. M-A. Knus and M. Ojanguren, Théorie de la Descente et Algèbres d’Azumaya, LNM 389, Springer Verlag, Berlin, 1974.

    Book  Google Scholar 

  63. J.-M. Kuo and G. Szeto, The structure of a partial Galois extension. II, J. Algebra Appl. 15, No. 4, Article ID 1650061, 12 p. (2016).

    Google Scholar 

  64. J.-M. Kuo and G. Szeto, Partial group actions and partial Galois extensions, Monatsh. Math (to appear).

    Google Scholar 

  65. A. R. Magid, The Separable Galois Theory of Commutative Rings, Marcel Dekker, 1974.

    MATH  Google Scholar 

  66. S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, LNM 818, Springer Verlag, 1980.

    Book  Google Scholar 

  67. S. Montgomery, Hopf Galois theory: A survey, Geo. Top. Monographs 16 (2009), 367–400.

    Article  MathSciNet  Google Scholar 

  68. S. Montgomery and D. S. Passman Outer Galois theory of prime rings, Rocky Mountain J. Math. 14 (1984), 305–318.

    Article  MathSciNet  Google Scholar 

  69. A. Masuoka and T. Yanai, Hopf module duality applied X-outer Galois theory, J. Algebra 265 (2003), 229–246.

    Article  MathSciNet  Google Scholar 

  70. P. J. McCarthy, Algebraic Extensions of Fields, Dover Pub., Inc., New York, 1991.

    MATH  Google Scholar 

  71. T. Nakayama, Galois theory for general rings with minimum condition, J. Math. Soc. Japan 1 (1949), 203–216.

    Article  MathSciNet  Google Scholar 

  72. T. Nakayama, Galois theory of simple rings, Trans. AMS 73 (1952), 276–292.

    Article  MathSciNet  Google Scholar 

  73. M. Ouyang, Azumaya extensions and Galois correspondence, Algebra Colloquium 7 (2000), 43–57.

    Article  MathSciNet  Google Scholar 

  74. A. Paques, Teoria de Galois Sobre Anillos Conmutativos, Univ. Los Andes, Mérida, Venezuela, 1999.

    Google Scholar 

  75. A. Paques, Teorias de Galois, Mini-course Notes (in portuguese), XXII Brazilian Algebra Meeting, UFBA, Brazil, 2012.

    Google Scholar 

  76. A. Paques, Galois, Morita and the trace map: a survey, São Paulo J. Math. Sc. 10 (2) (2016), 372–383.

    Article  Google Scholar 

  77. A. Paques, V. Rodrigues and A. Sant’Ana, Galois correspondences for partial Galois Azumaya extensions, J. Algebra Appl., 10, (5), (2011), 835–847.

    Article  MathSciNet  Google Scholar 

  78. A. Paques and A. Sant’Ana, When is a crossed product by a twisted partial action is Azumaya?, Comm. Algebra, 38, (2010), 1093–1103.

    Article  MathSciNet  Google Scholar 

  79. A. Paques and T. R. Tamusiunas, A Galois-Grothendieck-type correspondence for groupoid actions, Algebra Disc. Math. 17 (2014), 80–97.

    MathSciNet  MATH  Google Scholar 

  80. D. Passman Group Rings, Crossed Products and Galois Theory, CBMS 64, AMS, Providence, RI, 2000.

    Google Scholar 

  81. J. Rotman, Galois Theory, Springer-Verlag, New York, 1990.

    Book  Google Scholar 

  82. L. Rowen, Ring theory, Academic Press, Inc., Boston, MA, (1991)

    MATH  Google Scholar 

  83. I. Stewart, Galois Theory, Chapman and Hall, London, 1987.

    MATH  Google Scholar 

  84. G. Szeto and L. Xue, The Galois algebra with Galois group which is the automorphism group, J. Algebra 293 (2005), 312–318.

    Article  MathSciNet  Google Scholar 

  85. G. Szeto and L. Xue, On Galois algebras satisfying the fundamental theorem, Comm. Algebra 35 (2007), 3979–3985.

    Article  MathSciNet  Google Scholar 

  86. G. Szeto and L. Xue, On Galois extensions satisfying the fundamental theorem, Int. Math. Forum 36 (2007), 1773–1777.

    Article  MathSciNet  Google Scholar 

  87. G. Szeto and L. Xue, On Galois extensions with a one-to-one Galois map, Int. J. Algebra 5 (2011), 801–807.

    MathSciNet  MATH  Google Scholar 

  88. T. R. Tamusiunas, Teorias de Galois para ação de grupóides, Tese de Doutorado, UFRGS, Brasil, 2012.

    Google Scholar 

  89. K. H. Ulbrich, Galois erweiterungen von nicht-kommutativen ringen, Comm. Algebra 10 (1982), 655–672.

    Article  MathSciNet  Google Scholar 

  90. O. E. Villamayor and D. Zelinsky, Galois theory with finitely many idempotents, Nagoya math. j. 27 (1966), 721–731.

    Article  MathSciNet  Google Scholar 

  91. O. E. Villamayor and D. Zelinsky, Galois theory with infinitely many idempotents, Nagoya Math. J. 35 (1969), 83–98.

    Article  MathSciNet  Google Scholar 

  92. S. Westreich and T. Yanai, More about a Galois-type correspondence theory, J. Algebra 246 (2001), 629–640.

    Article  MathSciNet  Google Scholar 

  93. T. Yanai, Correspondences theory of Kharchenko and X-outer actions of pointed Hopf algebras, Comm. Algebra 25 (1997), 1713–1740.

    Article  MathSciNet  Google Scholar 

  94. T. Yanai, Galois correspondence theorem for Hopf algebra actions, Contemp. Math. 376 (2005), 393–411.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil

    Antonio Paques

Authors
  1. Antonio Paques
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Paques .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, São Paulo, Brazil

    Carlile Lavor

  2. Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, São Paulo, Brazil

    Francisco A. M. Gomes

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Paques, A. (2018). Galois Theories: A Survey. In: Lavor, C., Gomes, F. (eds) Advances in Mathematics and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-94015-1_11

Download citation

Publish with us

Policies and ethics

Search

Navigation