Exemplary Algebra – A Study of Algebra Through Examples
Authors
Description
Undoubtedly, Algebra is indispensable today to those in search of a solid mathematical education, just pay attention to the number of medals in the area to convince himself. This introductory book seeks to illustrate the theory as well as interactions with other areas (Geometry, Linear Algebra, Analysis, etc.) through more of 300 concrete examples. Great emphasis is placed on the hierarchy of the central ideas, in order to help the reader to differentiate the main results from the technical details. With this, the authors hope to convey their charm and enthusiasm for area. After all, Algebra should be the study of structures fundamental principles of Mathematics and not the “study of definitions”.
Target audience
Higher education
Name: Exemplary Algebra – A Study of Algebra Through Examples
Author(s): Sérgio Tadao Martins e Eduardo Tengan
Pages: 696
Publication: IMPA, 2020
ISBN: 978-65-89124-05-4
Edition: 1
Foreword
I Groups
1 Working in groups
1.1 Definitions
1.2 Early examples
1.3 Elemental properties
1.4 Translations and the “spin-turn”
1.5 Conjugation
1.6 Subgroups
1.7 Exercises
2 Cyclical order and groups
2.1 Definitions
2.2 Generators of a finite cyclic group
2.3 Exercises
3 Lagrange’s theorem
3.1 Side classes
3.2 Lagrange’s theorem
3.3 Exercises
4 Morphisms, Isomorphisms, and Automorphisms
4.1 Group morphisms
4.2 Isomorphisms
4.3 Automorphisms
4.4 Exercises
5 Groups that appear in the wild
5.1 Symmetric group
5.1.1 Parity of permutations and alternating group
5.2 Linear group
5.2.1 Elementary matrices
5.2.2 Orthogonal, unitary, and symplectic groups
5.3 Dihedral group
5.4 Free group
5.5 Fundamental group
5.5.1 Brouwer’s fixed-point theorem
5.5.2 The fundamental theorem of algebra
5.6 Other examples
5.6.1 Related group
5.6.2 Elliptic Curves
5.6.3 Thompson’s Group F
5.7 Exercises
6 Normal subgroups and quotients
6.1 Normal subgroups
6.2 Quotient group
6.3 Correspondence Theorem
6.4 Isomorphism Theorem
6.5 Exercises
7 Group action
7.1 Notation and definitions
7.2 Examples of actions
7.3 Partition and orbit-stabilizer theorems
7.4 P-groups
7.5 Burnside’s lemma
7.6 La liberte, la fidelité et la transitivité
7.7 Geometric actions
7.7.1 Projective space and the linear projective group
7.7.2 Actions on the hyperbolic plane
7.8 Exercises
8 Sorting Groups
8.1 Exact sequences
8.2 Semi-direct product
8.3 Presentation of a group
8.4 Simple groups
8.5 Sylow’s theorems
8.6 Soluble and nilpotent groups
8.7 Exercises
II Rings
9 Working on Rings
9.1 Basic Definitions and Properties
9.2 Units, nilpotent and idempotent
9.3 Domains, division rings, and bodies
9.4 Morphisms, isomorphisms, and automorphisms
9.5 Field of fractions
9.6 Exercises
10 Quotients
10.1 Ideals
10.2 Modules and algebras
10.3 Quotients
10.4 Isomorphism Theorem
10.5 Prime and Maximal Ideals
10.6 Chinese Remnants Theorem
10.7 Exercises
II Rings appearing in Nature
11.1 Polynomials
11.2 P-adistic integers
11.3 Algebraic sets
11.4 Group ring and representations
11.5 Exercises
12 Single Factorization Domains
12.1 Irreducible, cousins and associates
12.2 Single-factorization domains
12.3 Open issues
12.4 Exercises
13 Euclidean domains
13.1 Definition and examples
13.2 Single factorization in DE’s
13.3 Euclid’s algorithm
13.4 Example: Gauβ integers
13.5 Exercises
14 Noetherian Rings and Modules
14.1 Definition and examples
14.2 Hilbert’s basis theorem
14.3 Noetherian modules
14.4 Domains of Major Ideals
14.5 Exercises
15 Gauβ’s lemma
15.1 Gauβ’s lemma
15.2 Eisenstein’s criterion of irreducibility
15.3 Exercises
16 Finitely generated modules over DIP’s
16.1 Free bases and modules
16.2 Free modules on a DIP
16.3 Applications
16.4 Proofs of the principal theorems
16.5 Exercises
III Bodies
17 Basic Definitions
17.1 Characteristic of a group
17.2 Body extensions
17.3 Exercises
18 Algebraic and transcendent extensions
18.1 Definitions and examples
18.2 Minimal polynomial
18.3 Explicit transcendent elements
18.4 Exercises
19 Finite and Simple Extensions
19.1 Degree of an extension
19.2 Simple extensions
19.3 The motto of Degrees
19.4 Finite extensions are algebraic
19.5 Exercises
20 Constructions with ruler and compass
20.1 Constructible numbers
20.2 Constructability criterion
20.3 Exercises
21 Root body and algebraic closure
21.1 Root body of a polynomial
21.2 Algebraically closed bodies
21.3 Exercises
22 Immersions and automorphisms of body extensions
22.1 K-Immersions
22.2 The pickle principle and the fundamental motto
22.3 Immersions in algebraically closed bodies
22.4 Exercises
23 Galois theory: statement and examples
23.1 Galoisian extensions
23.2 The fundamental theorem of Galois’ theory
23.3 Early examples
23.4 GauB and the regular heptaecagon
23.5 The fundamental theorem of algebra revisited
23.6 Cyclic Extensions
23.7 Exercises
24 Normal and separable extensions
24.1 Separable polynomials
24.2 The Immersive Criterion of Separability
24.3 Normal extensions
24.4 Fundamental Theorem of Galois Theory
24.5 Open issues
24.6 Exercises
25 Finite Fields, Norm, and Trace
25.1 Finite Fields
25.2 Norm and Trace
25.3 Exercises
26 Solubility by radicals
26.1 Extensions of radical bodies and towers
26.2 Radical solubility criterion
26.3 Exercises
IV Appendices
The Fundamentals
A.1 Relationships
A.1.1 Relations of equivalence and order
A.1.2 Axiom of choice, Zorn’s lemma and Good Ordering
A.2 Divisibility and congruences
A.3 Exercises
B Complex Numbers
B.1 Polar form and Euler’s formula
B.2 Roots of unity
B.3 Exercises
C Vector spaces
C.1 Vector Spaces
C.2 Linear transformations
C.2.1 Matrix of a linear transformation with respect to a base
C.2.2 Eigenvalues and eigenvectors
C.3 Operations with vector spaces
C.3.1 Direct sum
C.3.2 Quotient
C.3.3 Dual
C.3.4 Tensor Product
C.3.5 External product
C.3.6 Determinant
C.4 Bilinear forms and quadratic forms
C.4.1 Bilinear forms and orthogonality
C.4.2 Symmetrical power
C.4.3 Domestic product
C.5 Exercises
Bibliography