Abstract Algebra: Suitable for Self-Study or Online Lectures

ISBN-13: 9783662679739 / Angielski / Miękka / 2024 / 332 str.

Marco Hien

Spis treści

1 Motivation and prerequisites

1.1 Goals

1.2 Prerequisites

2 Field extensions and algebraic elements

2.1 Field extensions

2.2 Intermediate Field and algebraic elements

Tasks

3 Groups

3.1 General Definition and Consequences

3.2 Subgroups and group homorphisms

Tasks

4 Group quotients and normal divisors

4.1 Equivalence relations

4.2 Group quotients

4.3 Lagrange's theorem

4.4 Normal Divisors and Factor Groups

4.5 The Homomorphism Theorem for Groups

4.6 Finite cyclic groups

Tasks

5 Rings and Ideals

5.1 Commutative rings with one

5.2 Ring homomorphisms

5.3 Units and zero divisors

5.4 Ideals, factor rings and the homomorphism theorem

5.5 Primideals and maximal ideals

5.6 The Chinese Remainder Theorem

5.7 Examples of rings in square number fields

Tasks

6 Euclidean rings, principal ideal rings, Noether's rings

6.1 Euclidean rings

6.2 The Euclidean algorithm

6.3 Noether's rings

Tasks

7 Factorial Rings

7.1 Prime elements and irreducible elements, factorial rings

7.2 Properties

Tasks

8 Quotient Fields for Integrity Ranges

Tasks

9 Irreducible polynomials in factorial rings

9.1 Content of polynomials

9.2 Reduction modulo Primelement

9.3 The Gauss Lemma

9.4 Application of the reduction mod ? and Gauss' theorem

Tasks 

10 Galois Theory (I) - Theorem A and its Variant A'

10.1 The miraculous Field creation

10.2 The decomposition Field 

10.3 Theorem A and A'

10.4 Application in the Field tower 

10.5 The Galois group

Tasks

11 Intermezzo - explicit example

Tasks

12 Normal Field extensions

12.1 Algebraic closure

12.2 Continuation of Field homorphisms

12.3 Normal extensions

Tasks

13 Separability 

13.1 Motivation and Definition

13.2 Formal derivation

13.3 Characteristics of a Field and separability

13.4 The degree of separability

13.5 The theorem of the primitive element

Tasks

14 Galois Theory (II) - The Main Theorem

14.1 The Main Theorem - Statement

14.2 Prospect of an application - midnight formula for all degrees?

14.3 Proof of the Main Theorem

14.4 Proof of the addition

Tasks

15 Cyclotomic fields 

15.1 Unit roots

15.2 Circle divisors and polynomials

Tasks 

16 Finite Fields

16.1 Prime fields, finite field and Frobenius

16.2 Finite Fields

Tasks

17 More Group Theory - Group Operations and Sylow

17.1 Group operations

17.2 The Sylow Theorems

17.3 Applications of the Sylow Theorems and Common Tricks

17.4 Proof of the Sylow Theorems

Tasks

18 Solvability of polynomial equations

18.1 Solvable groups

18.2 Solving polynomial equations by radicals

18.3 The general equation n-th degree

Tasks

A Proof of the existence of an algebraic closure 

B Tricks and methods to classify groups of a given order

B.1 Standard arguments and examples

B.2 Explicit examples

O autorze

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.

Blurb

This book contains the fundamental basics of algebra at university level. 

In addition to elementary algebraic structures such as groups, rings and fields, the text in particular covers Galois theory together with its applications to cyclotomic fields, finite fields as well as solving polynomial equations.

Special emphasis is placed on the natural development of the contents. Various supplementary explanations support this basic idea, point out connections and help to better comprehend the underlying concepts. 

The book is particularly suited as a textbook for learning algebra in self-study or to accompany online lectures.

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

The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.