Abstract
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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Notes
- 1.
Emil Artin, 3 March 1898–20 December 1962.
- 2.
Julius Wilhelm Richard Dedekind, 6 October 1831–12 February 1916.
- 3.
Pierre de Fermat, 17 August 1601–12 January 1665.
- 4.
Ernst Sigismund Fischer, 12 July 1875–14 November 1954.
- 5.
Emmy Noether, 23 March 1882–14 April 1935.
References
E. Artin, Galois Theory, Dover Publications, 1997.
E. Formanek, Rational function fields. Noether’s problem and related questions. Journal of Pure and Applied Algebra, 31(1984), 28–36.
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Brzeziński, J. (2018). Automorphism Groups of Fields. In: Galois Theory Through Exercises. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-72326-6_6
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DOI: https://doi.org/10.1007/978-3-319-72326-6_6
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