Automorphism Groups of Fields

  • Juliusz Brzeziński
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


In this chapter, we study automorphism groups of fields and introduce Galois groups of finite field extensions. The term “Galois group” is often reserved for automorphism groups of Galois field extensions, which we define and study in Chap.  9. The terminology used in this book is very common and has several advantages in textbooks (i.e. it is easier to formulate exercises). A central result of this chapter is Artin’s lemma, which is a key result in the modern presentation of Galois theory. In the exercises, we find Galois groups of many field extensions and we use also use this theorem for various problems on field extensions and their automorphism groups.


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    E. Artin, Galois Theory, Dover Publications, 1997.Google Scholar
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    E. Formanek, Rational function fields. Noether’s problem and related questions. Journal of Pure and Applied Algebra, 31(1984), 28–36.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Juliusz Brzeziński
    • 1
    • 2
  1. 1.Department of Mathematical SciencesUniversity of GothenburgGöteborgSweden
  2. 2.Chalmers University of TechnologyGöteborgSweden

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