Galois Theory Through Exercises pp 65-71 | Cite as

# Galois Modules

## Abstract

This chapter also illustrates the main theorems of Galois theory, though with more of a theoretical focus. We discuss some general notions and prove three important theorems that have several applications in algebra and number theory. These three theorems are related in one way or another to Galois modules, i.e. groups on which Galois groups of field extensions act as transformation groups. The first theorem is on the existence of so-called normal bases and concerns especially nice bases of finite Galois extensions. It is related to the Galois module structure on the additive group of the field. The second theorem, Hilbert’s theorem 90, is related to both additive and multiplicative structures of Galois extensions. This theorem has numerous applications which are discussed in the exercises. The third theorem is related to Hilbert’s theorem 90. Here we discuss so-called Kummer extensions. A particular case of such extensions are cyclic extensions. Hilbert’s theorem 90 may be considered in the context of so-called cohomology groups (defined for G-modules), which we mention briefly. We further classify all cubic and quartic Galois extensions of rational numbers in the context of cohomology groups.

## References

- [L]S. Lang, Algebra, Third Edition, Addison-Wesley, 1993.Google Scholar
- [S1]J.-P. Serre, Galois Cohomology, Springer, 1997.Google Scholar