A First Course in Abstract Algebra: Rings, Groups and Fields, (3rd Edition) by Marlow Anderson, Todd Feil
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A First Course in Abstract Algebra: Rings, Groups and Fields, (3rd Edition) written by
Marlow Anderson, Colorado College, Colorado Springs, USA and
Todd Feil, Denison University, Granville, Ohio, USA.
Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective. Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last 'of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals. Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book. As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.
A First Course in Abstract Algebra: Rings, Groups and Fields, (3rd Edition) by Marlow Anderson, Todd Feil written by
Joseph A. Gallian
cover the following topics.
I Numbers, Polynomials, and Factoring
1. The Natural Numbers
1.1 Operations on the Natural Numbers
1.2 Well Ordering and Mathematical Induction
1.3 The Fibonacci Sequence
1.4 Well Ordering Implies Mathematical Induction
1.5 The Axiomatic Method
2. The Integers
2.1 The Division Theorem
2.2 The Greatest Common Divisor
2.3 The GCD Identity
2.4 The Fundamental Theorem of Arithmetic
2.5 A Geometric Interpretation
3. Modular Arithmetic
3.1 Residue Classes
3.2 Arithmetic on the Residue Classes
3.3 Properties of Modular Arithmetic
4. Polynomials with Rational Coecients
4.1 Polynomials
4.2 The Algebra of Polynomials
4.3 The Analogy between Z and Q[x]
4.4 Factors of a Polynomial
4.5 Linear Factors
4.6 Greatest Common Divisors
5. Factorization of Polynomials
5.1 Factoring Polynomials
5.2 Unique Factorization
5.3 Polynomials with Integer Coecients
Section I in a Nutshell
II Rings, Domains, and Fields
6. Rings
6.1 Binary Operations
6.2 Rings
6.3 Arithmetic in a Ring
6.4 Notational Conventions
6.5 The Set of Integers Is a Ring
7. Subrings and Unity
7.1 Subrings
7.2 The Multiplicative Identity
7.3 Surjective, Injective, and Bijective Functions
7.4 Ring Isomorphisms
8. Integral Domains and Fields
8.1 Zero Divisors
8.2 Units
8.3 Associates
8.4 Fields
8.5 The Field of Complex Numbers
8.6 Finite Fields
9. Ideals
9.1 Principal Ideals
9.2 Ideals
9.3 Ideals That Are Not Principal
9.4 All Ideals in Z Are Principal
10. Polynomials over a Field
10.1 Polynomials with Coecients from an Arbitrary Field
10.2 Polynomials with Complex Coecients
10.3 Irreducibles in R[x]
10.4 Extraction of Square Roots in C
Section II in a Nutshell
III Ring Homomorphisms and Ideals
11. Ring Homomorphisms
11.1 Homomorphisms
11.2 Properties Preserved by Homomorphisms
11.3 More Examples
11.4 Making a Homomorphism Surjective
12. The Kernel
12.1 The Kernel
12.2 The Kernel Is an Ideal
12.3 All Pre-images Can Be Obtained from the Kernel
12.4 When Is the Kernel Trivial?
12.5 A Summary and Example
13. Rings of Cosets
13.1 The Ring of Cosets
13.2 The Natural Homomorphism
14. The Isomorphism Theorem for Rings
14.1 An Illustrative Example
14.2 The Fundamental Isomorphism Theorem
14.3 Examples
15. Maximal and Prime Ideals
15.1 Irreducibles
15.2 Maximal Ideals
15.3 Prime Ideals
15.4 An Extended Example
15.5 Finite Products of Domains
16. The Chinese Remainder Theorem
16.1 Some Examples
16.2 Chinese Remainder Theorem
16.3 A General Chinese Remainder Theorem
Section III in a Nutshell
V Groups
17. Symmetries of Geometric Figures
17.1 Symmetries of the Equilateral Triangle
17.2 Permutation Notation
17.3 Matrix Notation
17.4 Symmetries of the Square
17.5 Symmetries of Figures in Space
17.6 Symmetries of the Regular Tetrahedron
18. Permutations
18.1 Permutations
18.2 The Symmetric Groups
18.3 Cycles
18.4 Cycle Factorization of Permutations
19. Abstract Groups
19.1 De nition of Group
19.2 Examples of Groups
19.3 Multiplicative Groups
20. Subgroups
20.1 Arithmetic in an Abstract Group
20.2 Notation
20.3 Subgroups
20.4 Characterization of Subgroups
20.5 Group Isomorphisms
21. Cyclic Groups
21.1 The Order of an Element
21.2 Rule of Exponents
21.3 Cyclic Subgroups
21.4 Cyclic Groups
Section IV in a Nutshell
VI Group Homomorphisms
22. Group Homomorphisms
22.1 Homomorphisms
22.2 Examples
22.3 Structure Preserved by Homomorphisms
22.4 Direct Products
23. Structure and Representation
23.1 Characterizing Direct Products
23.2 Cayley's Theorem
24. Cosets and Lagrange's Theorem
24.1 Cosets
24.2 Lagrange's Theorem
24.3 Applications of Lagrange's Theorem
25. Groups of Cosets
25.1 Left Cosets
25.2 Normal Subgroups
25.3 Examples of Groups of Cosets
26. The Isomorphism Theorem for Groups
26.1 The Kernel
26.2 Cosets of the Kernel
26.3 The Fundamental Theorem
Section V in a Nutshell
VI Topics from Group Theory
27. The Alternating Groups
27.1 Transpositions
27.2 The Parity of a Permutation
27.3 The Alternating Groups
27.4 The Alternating Subgroup Is Normal
27.5 Simple Groups
28. Sylow Theory: The Preliminaries
28.1 p-groups
28.2 Groups Acting on Sets
29. Sylow Theory: The Theorems
29.1 The Sylow Theorems
29.2 Applications of the Sylow Theorems
29.3 The Fundamental Theorem for Finite Abelian Groups
30. Solvable Groups
30.1 Solvability
30.2 New Solvable Groups from Old
Section VI in a Nutshell
VII Unique Factorization
31. Quadratic Extensions of the Integers
31.1 Quadratic Extensions of the Integers
31.2 Units in Quadratic Extensions
31.3 Irreducibles in Quadratic Extensions
31.4 Factorization for Quadratic Extensions
32. Factorization
32.1 How Might Factorization Fail?
32.2 PIDs Have Unique Factorization
32.3 Primes
33. Unique Factorization
33.1 UFDs .
33.2 A Comparison between Z and Z[p??5]
33.3 All PIDs Are UFDs
34. Polynomials with Integer Coecients
34.1 The Proof That Q[x] Is a UFD
34.2 Factoring Integers out of Polynomials
34.3 The Content of a Polynomial
34.4 Irreducibles in Z[x] Are Prime
35. Euclidean Domains
35.1 Euclidean Domains
35.2 The Gaussian Integers
35.3 Euclidean Domains Are PIDs
35.4 Some PIDs Are Not Euclidean
Section VII in a Nutshell
VIII Constructibility Problems
36. Constructions with Compass and Straightedge
36.1 Construction Problems
36.2 Constructible Lengths and Numbers
37. Constructibility and Quadratic Field Extensions
37.1 Quadratic Field Extensions
37.2 Sequences of Quadratic Field Extensions
37.3 The Rational Plane
37.4 Planes of Constructible Numbers
37.5 The Constructible Number Theorem
38. The Impossibility of Certain Constructions
38.1 Doubling the Cube
38.2 Trisecting the Angle
38.3 Squaring the Circle
Section VIII in a Nutshell
IX Vector Spaces and Field Extensions
39. Vector Spaces I
39.1 Vectors
39.2 Vector Spaces
40. Vector Spaces II
40.1 Spanning Sets
40.2 A Basis for a Vector Space
40.3 Finding a Basis
40.4 Dimension of a Vector Space
41. Field Extensions and Kronecker's Theorem
41.1 Field Extensions
41.2 Kronecker
s Theorem
41.3 The Characteristic of a Field
42. Algebraic Field Extensions
42.1 The Minimal Polynomial for an Element
42.2 Simple Extensions
42.3 Simple Transcendental Extensions
42.4 Dimension of Algebraic Simple Extensions
43. Finite Extensions and Constructibility Revisited
43.1 Finite Extensions
43.2 Constructibility Problems
Section IX in a Nutshell
X Galois Theory
44. The Splitting Field
44.1 The Splitting Field
44.2 Fields with Characteristic Zero
45. Finite Fields
45.1 Existence and Uniqueness
45.2 Examples
46 Galois Groups
46.1 The Galois Group
46.2 Galois Groups of Splitting Fields
47. The Fundamental Theorem of Galois Theory
47.1 Subgroups and Sub elds
47.2 Symmetric Polynomials
47.3 The Fixed Field and Normal Extensions
47.4 The Fundamental Theorem
47.5 Examples
48. Solving Polynomials by Radicals
48.1 Field Extensions by Radicals
48.2 Re ning the Root Tower
48.3 Solvable Galois Groups
Section X in a Nutshell
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