Full Description
This book contains the basics of abstract algebra.
In addition to elementary algebraic structures such as groups, rings and solids, Galois theory in particular is developed together with its applications to the cyclotomic fields, finite fields or the question of the resolution of polynomial equations.
Special attention is paid to the natural development of the contents. Numerous intermediate explanations support this basic idea, show connections and help to better penetrate the underlying concepts.
The book is therefore particularly suitable for learning algebra in self-study or accompanying online lectures.
Contents
Motivation and prerequisites.- Field extensions and algebraic elements.- Groups.- Group quotients and normal divisors.- Rings and ideals.- Euclidean rings, principal ideal rings, Noetherian rings.- Factorial rings.- Quotient fields for integrality domains.- Irreducible polynomials in factorial rings.- Galois theory (I) - Theorem A and its variant A'.- Intermezzo - explicit example.- Normal fields extensions.- Separability.- Galois theory (II) - The main theorem.- Cyclotomic fields.- Finite fields.- More group theory - Group operations and Sylow.- Resolvability of polynomial equations.
