• Book Name: A First Course in Abstract Algebra Rings Groups and Fields
• Author: Marlow Anderson, Todd Feil
• Pages: 547
• Size: 3 MB

## Contents of A First Course in Abstract Algebra Pdf

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 Coefficients

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 Coefficients

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 Coefficients from an Arbitrary Field

10.2 Polynomials with Complex Coefficients

10.3 Irreducibles in R[x]

10.4 Extraction of Square Roots in C

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

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 207

19.1 Definition 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

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 283

26.1 The Kernel

26.2 Cosets of the Kernel

26.3 The Fundamental Theorem

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

Unique Factorization

31 Quadratic Extensions of the Integers

31.1 Quadratic Extensions of the Integers

32 Factorization

32.1 How Might Factorization Fail?

32.2 PIDs Have Unique Factorization

32.3 Primes

33 Unique Factorization 351

33.1 UFDs

33.2 A Comparison between Z and Z[p􀀀5]

33.3 All PIDs Are UFDs

34 Polynomials with Integer Coefficients

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 361

35.1 Euclidean Domains

35.2 The Gaussian Integers

35.3 Euclidean Domains Are PIDs

35.4 Some PIDs Are Not Euclidean

Constructibility Problems

36 Constructions with Compass and Straightedge

36.1 Construction Problems

36.2 Constructible Lengths and Numbers

37 Constructibility and Quadratic Field Extensions

37.2 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

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

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 Subfields

47.2 Symmetric Polynomials

47.3 The Fixed Field and Normal Extensions

47.4 The Fundamental Theorem

47.5 Examples