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## Über dieses Buch

This text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences.

Applications include:

Identification numbers and modular arithmetic(linear) error-correcting codes, including cyclic codesruler and compass constructionscryptographysymmetry of patterns in the real plane

Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.

## Inhaltsverzeichnis

### Chapter 1. Identification Numbers and Modular Arithmetic

Abstract
The first topic we will investigate is the mathematics of identification numbers. Many familiar things are described by a code of digits; zip codes, items in a grocery store, and books, to name three. One feature to all of these codes is the inclusion of an extra numerical digit, called a check digit, designed to detect errors in reading the code. When a machine (or a human) reads information, there is always the possibility of the information being read incorrectly. For example, moisture or dirt on the scanner used by a grocery store clerk can prevent an item’s code from being read correctly. It would be unacceptable if, because of a scanning error, customers were charged for caviar when they are buying tuna fish. The use of the check digit allows for the detection of some scanning errors. If an error is detected, the item is re-scanned until the correct code is read.
David R. Finston, Patrick J. Morandi

### Chapter 2. Error Correcting Codes

Abstract
The identification number schemes we discussed in the previous chapter give us the ability to determine if an error has been made in recording or transmitting information. However, they are limited in two ways. First, the types of errors detected are fairly restrictive, e.g. single digit errors or interchanging digits. Second, they provide no way to recover the intended information. Some more sophisticated ideas and mathematical concepts enable methods to encoding and transmit information in ways that allow both detection and correction of errors. There are many applications of these so-called error correcting codes, among them transmission of digital images from planetary probes and playing compact discs and DVD movies.
David R. Finston, Patrick J. Morandi

### Chapter 3. Rings and Fields

Abstract
We are all familiar with the natural, rational, real, and complex number systems and their arithmetic, but other mathematical systems exhibit similar arithmetic properties. The previous chapter, for instance, introduced the set of integers modulo n, and its addition, subtraction, and multiplication. In high school algebra you worked with polynomials, and saw how to add, subtract, and multiply them. In linear algebra you saw how arithmetic operations are performed on matrices, and might have seen vector spaces, with their addition, subtraction, and scalar multiplication. Many of the functions you studied in precalculus and calculus can be combined by addition, subtraction, multiplication, division, and also composition.
David R. Finston, Patrick J. Morandi

### Chapter 4. Linear Algebra and Linear Codes

Abstract
In this chapter we review the main ideas of linear algebra. The one twist is that we allow our scalars to come from any field instead of just the field of real numbers. In particular, the notion of a vector space over the field $$\mathbb{Z}_{2}$$ will be essential in our study of coding theory. We will also need to look at other finite fields when we discuss Reed–Solomon codes. One benefit to working with finite dimensional vector spaces over finite fields is that all sets in question are finite, and so computers can be useful in working with them.
David R. Finston, Patrick J. Morandi

### Chapter 5. Quotient Rings and Field Extensions

Abstract
In this chapter we describe a method for producing new rings from a given one. Of particular interest for applications is the case of a field extension of a given field.
David R. Finston, Patrick J. Morandi

### Chapter 6. Ruler and Compass Constructions

Abstract
One remarkable application of abstract algebra arises in connection with four classical questions, originating with mathematicians of ancient Greece, about geometric constructions.
David R. Finston, Patrick J. Morandi

### Chapter 7. Cyclic Codes

Abstract
In this chapter we will build codes from quotient rings of $$\mathbb{Z}_{2}[x]$$. One advantage of this construction will be that we can guarantee a certain degree of error correction. We first make a connection between words and elements of such a quotient ring.
David R. Finston, Patrick J. Morandi

### Chapter 8. Groups and Cryptography

Abstract
The final two applications of abstract algebra we will discuss are to cryptography, i.e., secure transmission of private information, and to the classification of geometric patterns in the plane $$\mathbb{R}^{2}$$. The algebraic structure at the heart of both applications is that of a group.
David R. Finston, Patrick J. Morandi

### Chapter 9. The Structure of Groups

Abstract
The application of group theory to cryptography discussed in the previous chapter utilized abelian groups, i.e., groups whose operation satisfies the commutative property. Nonabelian groups have also found application in many areas including cryptography, chemistry, physics, and even in interior and exterior decorating (wallpaper patterns and frieze patterns, respectively) as we’ll see in the final chapter of this text. The present chapter develops some general structure theory of groups essential to these applications.
David R. Finston, Patrick J. Morandi

### Chapter 10. Symmetry

Abstract
In this chapter we explore another connection between algebra and geometry. One of the main issues studied in plane geometry is congruence; roughly, two geometric figures are said to be congruent if one can be moved to coincide exactly with the other. We will be more precise below in our description of congruence, and investigating this notion will lead us to new examples of groups. The culmination of this discussion is the mathematical classification of frieze patterns and wallpaper patterns based on the structure of the groups that arise.
David R. Finston, Patrick J. Morandi

### Correction to: Identification Numbers and Modular Arithmetic

In the original version of Chapter 1, the figure was wrong which has now been replaced, at the end of the chapter opening page.
David R. Finston, Patrick J. Morandi

### Backmatter

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