SpringerLink
Log in

Galois Theory

  • Chapter
Algebra

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 211))

  • 186k Accesses

Abstract

This chapter contains the core of Galois theory. We study the group of automorphisms of a finite (and sometimes infinite) Galois extension at length, and give examples, such as cyclotomic extensions, abelian extensions, and even non-abelian ones, leading into the study of matrix representations of the Galois group and their classifications. We shall mention a number of fundamental unsolved problems, the most notable of which is whether given a finite group G, there exists a Galois extension of Q having this group as Galois group. Three surveys give recent points of view on those questions and sizeable bibliographies:

  • B. Matzat, Konstruktive Galoistheorie, Springer Lecture Notes 1284, 1987

  • B. Matzat, Ãœber das Umkehrproblem der Galoisschen Theorie, Jahrsbericht Deutsch. Mat.-Verein. 90 (1988), pp. 155–183

  • J. P. Serre, Topics in Galois theory, course at Harvard, 1989, Jones and Bartlett, Boston 1992 More specific references will be given in the text at the appropriate moment concerning this problem and the problem of determining Galois groups over specific fields , especially the rational numbers.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
USD 9.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 79.95
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. G. Belyi, Galois extensions of the maximal cyclotomic field, Izv. Akad. Nauk SSR 43 (1979) pp. 267–276 (= Math. USSR Izv. 14 (1980), pp. 247–256

    MathSciNet  Google Scholar 

  2. G. Belyi, On extensions of the maximal cyclotomic field having a given classical Galois group, J. reine angew. Math. 341 (1983), pp. 147–156

    MathSciNet  Google Scholar 

  3. A. Douady, Determination d’un groupe de Galois, C.R. Acad. Sci. 258 (1964), pp. 5305–5308

    MathSciNet  Google Scholar 

  4. S. Lang, Fundamentals of Diophantine Geometry. Springer Verlag 1983

    Google Scholar 

  5. F. Pop, Etale Galois covers of affine smooth curves, Invent. Math. 120 (1995), pp. 555–578

    Article  MathSciNet  Google Scholar 

  6. J.-P. Serre, Groupes de Galois sur Q, Séminaire Bourbaki, 1987–1988 Astérisque 161–162, pp. 73–85

    MathSciNet  Google Scholar 

  7. R.-Y. Shih, On the construction of Galois extensions of function fields and number fields, Math. Ann. 207 (1974), pp. 99–120

    Article  MathSciNet  Google Scholar 

  8. R. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), pp. 148–158

    Article  MathSciNet  Google Scholar 

  9. R. Swan, Noether’s problem in Galois theory, Emmy Noether in Bryn Mawr, J. D. Sally and B. Srinivasan, eds., Springer Verlag, 1983, pp. 40

    Google Scholar 

References

  1. K. Iwasawa, On solvable extension of algebraic number fields, Ann. of Math. 58 (1953), pp. 548–572

    Article  MathSciNet  Google Scholar 

  2. I. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR 18 (1954), pp. 525–578 (Amer. Math. Soc. Transl. 4 (1956), pp. 185–237)

    MathSciNet  Google Scholar 

Bibliography

  1. E. Witt, Der Existenzsatz für abelsche Funktionenkörper, J. reine angew. Math. 173 (1935), pp. 43–51.

    Article  MathSciNet  Google Scholar 

  2. E. Witt, Konstruktion von galoisschen körpern der Charakteristik p it vorgegebener Gruppe der Ordung p f, J. reine angew. Math. 174 (1936), pp. 237–245

    Article  MathSciNet  Google Scholar 

  3. E. Witt, Zyklische Körper und Algebren der Charakteristik p vom Grad p u. Sturkture diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p. J. reine angew. Math. 176 (1937), pp. 126–140.

    Article  MathSciNet  Google Scholar 

Bibliography

  1. S. Lang, Cyclotomic Fields I and II, Second Edition, Springer Verlag, 1990 (Combined edition from the first editions, 1978, 1980)

    Google Scholar 

  2. S. Lang and H. Trotter, Distribution of Frobenius Elements in GL2-Extensions of the Rational Numbers, Springer Lecture Notes 504 (1975)

    Google Scholar 

  3. J.-P. Serre, Abelian l-adic Representations and Elliptic Curves, Benjamin, 1968

    Google Scholar 

  4. J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), pp. 259–331

    Article  MathSciNet  Google Scholar 

  5. G. Shimura, Introduction to the arithmetic theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, 1971

    Google Scholar 

  6. K. Wingberg, On Galois groups of p-closed algebraic number fields with restricted ramification, I, J. reine angew. Math. 400 (1989), pp. 185–202; and II, ibid., 416 (1991), pp. 187–194

    MathSciNet  Google Scholar 

Bibliography

  1. E. Artin and J. Tate, Class Field Theory, Benjamin-Addison-Wesley, 1968 (reprinted by Addison-Wesley, 1991)

    Google Scholar 

  2. P. Deligne, Formes modulaires et représentations l-adiques, Séminaire Bourbaki 1968–1969, exp. No. 355

    Google Scholar 

  3. P. Deligne, Formes modulaires et représentations de GL(2), Springer Lecture Notes 349 (1973), pp. 55–105

    Google Scholar 

  4. P. Deligne and J. P. Serre, Formes modulaires de poids 1, Ann. Sci. ENS 7 (1974), pp. 507–530

    MathSciNet  Google Scholar 

  5. S. Lang, Algebraic Number Theory, Springer Verlag, reprinted from Addison-Wesley (1970)

    Google Scholar 

  6. S. Lang, Elliptic functions, Springer Verlag, 1973

    Google Scholar 

  7. S. Lang, Introduction to modular forms, Springer Verlag, 1976

    Google Scholar 

  8. K. Ribet, On modular representations of Gal(Ō/G) arising from modular forms, Invent. Math. 100 (1990), pp. 431–476

    Article  MathSciNet  Google Scholar 

  9. K. Ribet, From the Taniyama-Shimura conjecture to Fermat’s last theorem, Annales de la Fac. des Sci. Toulouse (1990), pp. 116–139

    Google Scholar 

  10. J.-P. Serre, Une interprétation des congruences relatives à la fonction de Ramanujan, Séminaire Delange-Pisot-Poitou, 1967–1968

    Google Scholar 

  11. J.-P. Serre, Congruences et formes modulaires (d’après Swinnerton-Dyer), Séminaire Bourbaki, 1971–1972

    Google Scholar 

  12. J.-P. Serre, A course in arithmetic, Springer Verlag, 1973

    Google Scholar 

  13. J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Ō/Q), Duke Math. J. 54 (1987), pp. 179–230

    Article  MathSciNet  Google Scholar 

  14. G. Shimura, A reciprocity law in non-solvable extensions, J. reine angew. Math. 221 (1966), pp. 209–220

    MathSciNet  Google Scholar 

  15. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten and Princeton University Press, 1971

    Google Scholar 

  16. H. P. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, (Antwerp conference) Springer Lecture Notes 350 (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 96520, USA

    Serge Lang

Authors
  1. Serge Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lang, S. (2002). Galois Theory. In: Algebra. Graduate Texts in Mathematics, vol 211. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0041-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0041-0_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6551-1

  • Online ISBN: 978-1-4613-0041-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation