Galois Theory(D)
GALOIS THEORY (D)
Groups, Rings and Modules is essential.
- Field extensions, tower law, algebraic extensions; irreducible polynomials and relation with simple algebraic extensions. Finite multiplicative subgroups of a field are cyclic. Existence and uniqueness of splitting fields. [6]
- Existence and uniquness of algebraic closure. [1]
- Separability. Theorem of primitive element. Trace and norm. [3] Normal and Galois extensions, automorphic groups. Fundamental theorem of Galois theory. [3]
- Galois theory of finite fields. Reduction mod p. [2]
- Cyclotomic polynomials, Kummer theory, cyclic extensions. Symmetric functions. Galois theory of cubics and quartics. [4]
- Solubility by radicals. Insolubility of general quintic equations and other classical problems. [3]
- Artin’s theorem on the subfield fixed by a finite group of automorphisms. Polynomial invariants of a finite group; examples. [2]
Appropriate books
- E. Artin Galois Theory. Dover Publications I. Stewart Galois Theory. Taylor & Francis Ltd Chapman & Hall/CRC 3rd edition
- B. L. van der Waerden Modern Algebra. Ungar Pub 1949
- S. Lang Algebra (Graduate Texts in Mathematics). Springer-Verlag New York Inc
- I. Kaplansky Fields and Rings. The University of Chicago Press
Associated GitHub Page
https://jaircambridge.github.io/Galois-Theory-D/
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