Weitere Kapitel dieses Buchs durch Wischen aufrufen
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.
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David Hilbert, 23 January 1862–14 February 1943.
Ernst Eduard Kummer, 29 January 1810–14 May 1893.
This creates a conflict with the historically older use of the notion of a normal polynomial, which we introduced in Exercise 7.9. Usually it is possible to easily recognize which notion is being discussed.
Otto Schreier, 3 March 1901–2 June 1929.
Zurück zum Zitat S. Lang, Algebra, Third Edition, Addison-Wesley, 1993. S. Lang, Algebra, Third Edition, Addison-Wesley, 1993.
Zurück zum Zitat J.-P. Serre, Galois Cohomology, Springer, 1997. J.-P. Serre, Galois Cohomology, Springer, 1997.
- Galois Modules
- Chapter 11
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