Galois Theory Through Exercises pp 35-41 | Cite as

# Automorphism Groups of Fields

## 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.

## References

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