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## Über dieses Buch

This textbook offers an innovative approach to abstract algebra, based on a unified treatment of similar concepts across different algebraic structures. This makes it possible to express the main ideas of algebra more clearly and to avoid unnecessary repetition.

The book consists of two parts: The Language of Algebra and Algebra in Action. The unified approach to different algebraic structures is a primary feature of the first part, which discusses the basic notions of algebra at an elementary level. The second part is mathematically more complex, covering topics such as the Sylow theorems, modules over principal ideal domains, and Galois theory.

Intended for an undergraduate course or for self-study, the book is written in a readable, conversational style, is rich in examples, and contains over 700 carefully selected exercises.

## Inhaltsverzeichnis

### Chapter 1. Glossary of Basic Algebraic Structures

Abstract
In this first chapter, we will introduce the algebraic structures that are the subject of this book (groups, rings, etc.), along with some accompanying fundamental notions. Elementary properties that follow easily from the definitions will be considered, and simple, basic examples will be given to help the reader understand and memorize the concepts. We will not go further than that. Our first “date” with algebraic structures will thus be rather formal and superficial, just to see what we will be dealing with.
Matej Brešar

### Chapter 2. Examples of Groups and Rings

Abstract
This chapter is a survey of the most prominent examples of groups and rings. Roughly speaking, the first half of the chapter is devoted to commutative structures and the second half to noncommutative structures. A ring is, in particular, an additive group, and its invertible elements form a multiplicative group. Examples of rings and groups will therefore be intertwined. One of the goals of the chapter is to show that groups and rings occur throughout mathematics. Exposure to algebraic ideas is therefore useful for mathematicians working in all areas.
Matej Brešar

### Chapter 3. Homomorphisms

Abstract
It often happens in mathematics that we understand something intuitively, but do not know how to express it in words. We then need appropriate definitions that help us to clarify our thoughts and make discussion possible. The “right” definition should also reveal the essence of the issue and direct our way of thinking. One of the definitions that plays such a role in algebra is that of a homomorphism. A homomorphism is a map from an algebraic structure to an algebraic structure of the same type that preserves the operation(s) of the structure. It is impossible to list all reasons why homomorphisms are important, they just arise naturally in various contexts. In particular, we need them for recognizing common features of algebraic structures.
Matej Brešar

### Chapter 4. Quotient Structures

Abstract
In this chapter, we will get acquainted with special types of subgroups called normal subgroups. With the help of a normal subgroup we can construct a new group called a quotient group. In a similar fashion, we construct quotient rings from certain subsets of rings called ideals. We have already encountered a particular case of these constructions in Section 2.2 on integers modulo n, but at that time we avoided the terms “quotient group” and “quotient ring”. Quotient structures are intimately connected with homomorphisms. Every homomorphism gives rise to a quotient structure, and every quotient structure gives rise to a homomorphism. Through this connection, we will better understand the meaning of homomorphisms that are not injective. However, the real meaning and importance of both homomorphisms and quotient structures will become evident in Part II, where we will use them as tools for solving various problems.
Matej Brešar

### Chapter 5. Commutative Rings

Abstract
We are at a new beginning. If the goal of Part I was gaining familiarity with algebraic structures, we now wish to understand them at a deeper level. This first chapter of Part II is primarily devoted to commutative rings that are in certain ways similar to the ring of integers. The most prominent example is $$F[X]$$, the ring of polynomials over a field F. We will see that the theory of divisibility of integers, developed in Section 2.1, holds in essentially the same form not only for $$F[X]$$, but for more general rings called principal integral domains. In the final part of the chapter, we will establish a theorem on modules over principal ideal domains, along with two surprising applications: one to group theory and the other to matrix theory.
Matej Brešar

### Chapter 6. Finite Groups

Abstract
The success in classifying finite Abelian groups in the preceding chapter should not mislead us. The theory of finite groups is extremely complex and answers to general questions do not come easy. It would be far too much to expect that the structure of arbitrary finite groups can be completely determined. What we will show in this chapter is that every finite group contains certain subgroups that are relatively well understood. Thus, instead of tackling the whole group, which would be too ambitious, we will focus on its smaller pieces.
Matej Brešar

### Chapter 7. Field Extensions

Abstract
This last chapter is the closest to the origin of algebra. Field extensions are intimately connected with polynomial equations, the subject of investigation of classical algebra. In the first section, we present a historical overview which should help the reader to better understand the content of this chapter, and, in fact, of the whole book. The theory developed in subsequent sections is mathematically rich with several impressive results, culminating in Galois theory, which, along with its applications, often strikes mathematical souls with its harmony and perfection.
Matej Brešar

### Backmatter

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