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Lee - Abstract Algebra: An Introductory Course

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Lee Abstract Algebra: An Introductory Course
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    Abstract Algebra: An Introductory Course
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This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.;Part I Preliminaries -- 1 Relations and Functions -- 2 The Integers and Modular Arithmetic -- Part II Groups -- 3 Introduction to Groups -- 4 Factor Groups and Homomorphisms -- 5 Direct Products and the Classification of Finite Abelian Groups -- 6 Symmetric and Alternating Groups -- 7 The Sylow Theorems -- Part III Rings -- 8 Introduction to Rings -- 9 Ideals, Factor Rings and Homomorphisms -- 10 Special Types of Domains -- Part IV Fields and Polynomials -- 11 Irreducible Polynomials -- 12 Vector Spaces and Field Extensions -- Part V Applications -- 13 Public Key Cryptography -- 14 Straightedge and Compass Constructions -- A The Complex Numbers -- B Matrix Algebra -- Solutions -- Index.

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Part I
Preliminaries
Springer International Publishing AG, part of Springer Nature 2018
Gregory T. Lee Abstract Algebra Springer Undergraduate Mathematics Series
1. Relations and Functions
Gregory T. Lee 1
(1)
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada
Gregory T. Lee
Email:
We begin by introducing some basic notation and terminology. Then we discuss relations and, in particular, equivalence relations, which we shall see several times throughout the book. In the final section, we talk about various sorts of functions.
1.1 Sets and Set Operations
A set is a collection of objects. We will see many sorts of sets throughout this course. Perhaps the most common will be sets of numbers. For instance, we have the set of natural numbers ,
Abstract Algebra An Introductory Course - image 1
the set of integers ,
and the set of rational numbers We also write for the set of real numbers and - photo 2
and the set of rational numbers
We also write for the set of real numbers and for the set of complex numbers - photo 3
We also write Picture 4 for the set of real numbers and Picture 5 for the set of complex numbers .
But sets do not necessarily consist of numbers. Indeed, we can consider the set of all letters of the alphabet, the set of all polynomials with even integers as coefficients or the set of all lines in the plane with positive slope.
The objects in a set are called its elements . We write Picture 6 if a is an element of a set S . Thus, Picture 7 but Picture 8 . The set with no elements is called the empty set , and denoted Picture 9 . Any other set is said to be nonempty .
If S and T are sets, then we say that S is a subset of T , and write Picture 10 , if every element of S is also an element of T . Of course, Picture 11 . We say that S is a proper subset of T , and write Picture 12 , if Picture 13 but Picture 14 . Thus, it is certainly true that Picture 15 , but we can be more precise and write Picture 16 .
For any two sets S and T , their intersection , Abstract Algebra An Introductory Course - image 17 , is the set of all elements that lie in S and T simultaneously.
Example 1.1.
Let Abstract Algebra An Introductory Course - image 18 and Abstract Algebra An Introductory Course - image 19 . Then Abstract Algebra An Introductory Course - image 20 .
We can extend this notion to the intersection of an arbitrary collection of sets. If I is a nonempty set and, for each Picture 21 , we have a set Picture 22 , then we write Picture 23 for the set of elements that lie in all of the Abstract Algebra An Introductory Course - image 24 simultaneously.
Example 1.2.
For each Abstract Algebra An Introductory Course - image 25 , let Abstract Algebra An Introductory Course - image 26 . Then Also for any sets S and T their union is the set of all elements - photo 27 .
Also, for any sets S and T , their union , is the set of all elements that lie in S or T or both Example 13 Using - photo 28 , is the set of all elements that lie in S or T (or both).
Example 1.3.
Using the same S and T as in Example , we have
Furthermore if I is a nonempty set and we have a set for each then we write - photo 29
Furthermore, if I is a nonempty set and we have a set Picture 30 for each Picture 31 , then we write Picture 32 for the union of all of the Picture 33 ; that is, the set of all elements that lie in at least one of the Abstract Algebra An Introductory Course - image 34 .
Example 1.4.
If we use the same sets Abstract Algebra An Introductory Course - image 35 as in Example , we have Abstract Algebra An Introductory Course - image 36 .
In addition, for any two sets S and T , the set difference (or relative complement ) is the set Abstract Algebra An Introductory Course - image 37 .
Example 1.5.
Once again using S and T as in Example , we have Abstract Algebra An Introductory Course - image 38 .
We will need one more definition. The following construction is named after Ren Descartes .
Definition 1.1.
Let S and T be any sets. Then the Cartesian product Picture 39
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