Algebra Notes from the Underground 1st Edition by Paolo Aluffi ISBN 1108958230 978-1108958233

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ISBN 10: 1108958230
ISBN 13: 978-1108958233
Author: Paolo Aluffi

From rings to modules to groups to fields, this undergraduate introduction to abstract algebra follows an unconventional path. The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics – such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory – are all treated in detail. Students will appreciate the text’s conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic logic and naïve set theory.

Algebra Notes from the Underground 1st Table of contents:

Part I Rings

1 The Integers

1.1 The Well-Ordering Principle and Induction

1.2 ‘Division with Remainder’ in Z

1.3 Greatest Common Divisors

1.4 The Fundamental Theorem of Arithmetic

2 Modular Arithmetic

2.1 Equivalence Relations and Quotients

2.2 Congruence mod n

2.3 Algebra in Z/nZ

2.4 Properties of the Operations +, · on Z/nZ

2.5 Fermat’s Little Theorem, and the RSA Encryption System

3 Rings

3.1 Definition and Examples

3.2 Basic Properties

3.3 Special Types of Rings

4 The Category of Rings

4.1 Cartesian Products

4.2 Subrings

4.3 Ring Homomorphisms

4.4 Isomorphisms of Rings

5 Canonical Decomposition, Quotients, and Isomorphism Theorems

5.1 Rings: Canonical Decomposition, I

5.2 Kernels and Ideals

5.3 Quotient Rings

5.4 Rings: Canonical Decomposition, II

5.5 The First Isomorphism Theorem

5.6 The Chinese Remainder Theorem

5.7 The Third Isomorphism Theorem

6 Integral Domains

6.1 Prime and Maximal Ideals

6.2 Primes and Irreducibles

6.3 Euclidean Domains and PIDs

6.4 PIDs and UFDs

6.5 The Field of Fractions of an Integral Domain

7 Polynomial Rings and Factorization

7.1 Fermat’s Last Theorem for Polynomials

7.2 The Polynomial Ring with Coefficients in a Field

7.3 Irreducibility in Polynomial Rings

7.4 Irreducibility in Q[x] and Z[x]

7.5 Irreducibility Tests in Z[x]

Part II Modules

8 Modules and Abelian Groups

8.1 Vector Spaces and Ideals, Revisited

8.2 The Category of R-Modules

8.3 Submodules, Direct Sums

8.4 Canonical Decomposition and Quotients

8.5 Isomorphism Theorems

9 Modules over Integral Domains

9.1 Free Modules

9.2 Modules from Matrices

9.3 Finitely Generated vs. Finitely Presented

9.4 Vector Spaces are Free Modules

9.5 Finitely Generated Modules over Euclidean Domains

9.6 Linear Transformations and Modules over k[t]

10 Abelian Groups

10.1 The Category of Abelian Groups

10.2 Cyclic Groups, and Orders of Elements

10.3 The Classification Theorem

10.4 Fermat’s Theorem on Sums of Squares

Part III Groups

11 Groups—Preliminaries

11.1 Groups and their Category

11.2 Why Groups? Actions of a Group

11.3 Cyclic, Dihedral, Symmetric, Free Groups

11.4 Canonical Decomposition, Normality, and Quotients

11.5 Isomorphism Theorems

12 Basic Results on Finite Groups

12.1 The Index of a Subgroup, and Lagrange’s Theorem

12.2 Stabilizers and the Class Equation

12.3 Classification and Simplicity

12.4 Sylow’s Theorems: Statements, Applications

12.5 Sylow’s Theorems: Proofs

12.6 Simplicity of A[sub(n)]

12.7 Solvable Groups

Part IV Fields

13 Field Extensions

13.1 Fields and Homomorphisms of Fields

13.2 Finite Extensions and the Degree of an Extension

13.3 Simple Extensions

13.4 Algebraic Extensions

13.5 Application: ‘Geometric Impossibilities’

14 Normal and Separable Extensions, and Splitting Fields

14.1 Simple Extensions, Again

14.2 Splitting Fields

14.3 Normal Extensions

14.4 Separable Extensions; and Simple Extensions Once Again

14.5 Application: Finite Fields

15 Galois Theory

15.1 Galois Groups and Galois Extensions

15.2 Characterization of Galois Extensions

15.3 The Fundamental Theorem of Galois Theory

15.4 Galois Groups of Polynomials

15.5 Solving Polynomial Equations by Radicals

15.6 Other Applications

Appendix A Background

A.1 Sets—Basic Notation

A.2 Logic

A.3 Quantifiers

A.4 Types of Proof: Direct, Contradiction, Contrapositive

A.5 Set Operations

A.6 Functions

A.7 Injective/Surjective/Bijective Functions

A.8 Relations; Equivalence Relations and Quotient Sets

A.9 Universal Property of Quotients and Canonical Decomposition

Appendix B Solutions to Selected Exercises

B.1 The Integers

B.2 Modular Arithmetic

B.3 Rings

B.4 The Category of Rings

B.5 Canonical Decomposition, Quotients, and Isomorphism Theorems

B.6 Integral Domains

B.7 Polynomial Rings and Factorization

B.8 Modules and Abelian Groups

B.9 Modules over Integral Domains

B.10 Abelian Groups

B.11 Groups—Preliminaries

B.12 Basic Results on Finite Groups

B.13 Field Extensions

B.14 Normal and Separable Extensions, and Splitting Fields

B.15 Galois Theory

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