Hien Abstract Algebra

Suitable for Self-Study or Online Lectures
1. Auflage 2024
ISBN: 978-3-662-67974-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 7, 307 Seiten, eBook

Reihe: Mathematics Study Resources

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.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Hien, Marco

Inhaltsverzeichnis

1 Motivation and prerequisites1.1 Goals1.2 Prerequisites2 Field extensions and algebraic elements2.1 Field extensions2.2 Intermediate Field and algebraic elementsTasks3 Groups3.1 General Definition and Consequences3.2 Subgroups and group homorphismsTasks4 Group quotients and normal divisors4.1 Equivalence relations4.2 Group quotients4.3 Lagrange's theorem4.4 Normal Divisors and Factor Groups4.5 The Homomorphism Theorem for Groups4.6 Finite cyclic groupsTasks5 Rings and Ideals5.1 Commutative rings with one5.2 Ring homomorphisms5.3 Units and zero divisors5.4 Ideals, factor rings and the homomorphism theorem5.5 Primideals and maximal ideals5.6 The Chinese Remainder Theorem5.7 Examples of rings in square number fieldsTasks6 Euclidean rings, principal ideal rings, Noether's rings6.1 Euclidean rings6.2 The Euclidean algorithm6.3 Noether's ringsTasks7 Factorial Rings7.1 Prime elements and irreducible elements, factorial rings7.2 PropertiesTasks8 Quotient Fields for Integrity RangesTasks9 Irreducible polynomials in factorial rings9.1 Content of polynomials9.2 Reduction modulo Primelement9.3 The Gauss Lemma9.4 Application of the reduction mod ? and Gauss' theoremTasks 10 Galois Theory (I) - Theorem A and its Variant A'10.1 The miraculous Field creation10.2 The decomposition Field 10.3 Theorem A and A'10.4 Application in the Field tower 10.5 The Galois groupTasks
11 Intermezzo - explicit example
Tasks12 Normal Field extensions12.1 Algebraic closure12.2 Continuation of Field homorphisms12.3 Normal extensionsTasks13 Separability 13.1 Motivation and Definition13.2 Formal derivation13.3 Characteristics of a Field and separability13.4 The degree of separability13.5 The theorem of the primitive elementTasks14 Galois Theory (II) - The Main Theorem14.1 The Main Theorem - Statement14.2 Prospect of an application - midnight formula for all degrees?14.3 Proof of the Main Theorem14.4 Proof of the additionTasks15 Cyclotomic fields 15.1 Unit roots15.2 Circle divisors and polynomialsTasks 16 Finite Fields16.1 Prime fields, finite field and Frobenius16.2 Finite FieldsTasks17 More Group Theory - Group Operations and Sylow17.1 Group operations17.2 The Sylow Theorems17.3 Applications of the Sylow Theorems and Common Tricks17.4 Proof of the Sylow TheoremsTasks18 Solvability of polynomial equations18.1 Solvable groups18.2 Solving polynomial equations by radicals18.3 The general equation n-th degreeTasksA Proof of the existence of an algebraic closure B Tricks and methods to classify groups of a given orderB.1 Standard arguments and examplesB.2 Explicit examples

Über Autor(innen)

After a postdoctoral year at the University of Chicago, Prof. Dr. Marco Hien initially worked at the University of Regensburg. Since 2010, he has been Professor of Algebra and Number Theory at the University of Augsburg with the research areas of algebraic geometry and algebraic analysis. In 2020, he received the "Prize for Good Teaching" from the Bavarian Ministry of Science.