Algebra — Basic
An Introduction to Abstract Algebra: Groups, Rings, and Fields (Undergraduate Level)
Overview
The basic level introduces the foundations of abstract algebra (modern algebra). We generalize number structures and study algebraic structures such as groups, rings, and fields.
Learning Objectives
- Understand the definition and basic properties of groups
- Develop intuition for group theory through concrete examples
- Understand the concepts of rings and fields
- Learn the structure of polynomial rings
- Learn the basics of field extensions
Table of Contents (18 chapters)
Introduction
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Chapter 1
Abstract Algebra
Overview of groups, rings, and fields
Group Theory
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Chapter 2
Monoid
Associative binary operation with identity
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Chapter 3
Groups
Definition of groups, subgroups, permutation groups
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Chapter 4
Abelian Groups
Groups with the commutative law
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Chapter 5
Cyclic Groups
Groups generated by a single element
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Chapter 6
Cosets and Normal Subgroups
Cosets, normal subgroups, and the kernel correspondence
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Chapter 7
Quotient Groups
Partitioning groups by normal subgroups and the isomorphism theorems
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Chapter 8
Lagrange's Theorem
The order of a subgroup divides the order of the group
Rings, Modules, and Fields
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Chapter 9
Rings, Modules, and Fields
Definition of rings, integral domains, fields, ideals, field of fractions, and modules
Polynomials
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Chapter 10
Polynomials
Definition, operations, roots, special polynomials, interpolation, applications
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Chapter 11
Binomial Theorem
Binomial expansion and binomial coefficients
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Chapter 12
Partial Fraction Decomposition
Decomposing rational functions into sums of simpler fractions
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Chapter 13
Discriminant
Discriminants of quadratic, cubic, and general-degree polynomials
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Chapter 14
Resultant
Sylvester matrix, product-of-roots formula, and the discriminant relation
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Chapter 15
Synthetic Division and Horner's Method
Efficient polynomial evaluation algorithms
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Chapter 16
Descartes' Rule of Signs
Upper bound for the number of real roots of a polynomial
History
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Chapter 17
History of Polynomials
Four millennia from ancient Egypt to modern Galois theory
Practice
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Chapter 18
Exercises
Comprehensive practice problems for the basic level
Prerequisites
- Introductory Algebra material (complex numbers, higher-degree equations)
- Basic concepts of sets and mappings
- Foundations of logical proof
Algebraic Structure of "Sum" and "Product"
Here we survey the two operations at the heart of algebra — "sum" and "product" — examining their common properties and mutual relationship.
Properties of "Sum"
The common properties of operations called "sum" are associativity and commutativity.
Basic Properties of Sum
- Associativity: \((X + Y) + Z = X + (Y + Z)\)
- Commutativity: \(X + Y = Y + X\)
- Identity element: there exists \(0\) with \(X + 0 = X\)
- Inverse element: there exists \(-X\) with \(X + (-X) = 0\)
Important difference from product: a sum is only defined between objects of the same type.
- vector + vector = vector ✓
- vector + scalar = ? ✗ (not defined)
- By contrast, a product can mix types: scalar × vector = vector
Examples of various "sums":
- Addition of numbers: \(3 + 5 = 8\), \((-2) + 7 = 5\)
- Sum of vectors: $\begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}$
- Sum of matrices: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$
- Sum of functions: \((f+g)(x) = f(x) + g(x)\)
- Direct sum of sets: \(A \oplus B\) (direct sum of algebraic structures)
- Exclusive OR: XOR (addition in \(\mathbb{Z}/2\mathbb{Z}\))
Properties of "Product"
What operations called "product" share is bilinearity.
What is Bilinearity?
A map \(f: V \times V \to W\) is bilinear if it is linear in each argument:
- \(f(aX + bY, Z) = a \cdot f(X, Z) + b \cdot f(Y, Z)\)
- \(f(Z, aX + bY) = a \cdot f(Z, X) + b \cdot f(Z, Y)\)
Comparison of major "products":
| Operation | Bilinear | Associative | Commutative | Notes |
|---|---|---|---|---|
| Ordinary product \(xy\) | ○ | ○ | ○ | |
| Matrix product \(AB\) | ○ | ○ | ✗ | |
| Inner product \(\langle X, Y \rangle\) | ○ | — | ○ | returns a scalar |
| Cross product \(X \times Y\) | ○ | ✗ | ✗ | anti-commutative |
| Lie bracket \([X, Y]\) | ○ | ✗ | ✗ | anti-commutative + Jacobi identity |
| Tensor product \(X \otimes Y\) | ○ | ○ | ✗ | dimensions multiply |
| Wedge product \(\omega \wedge \eta\) | ○ | ○ | ✗ | product of differential forms |
Associativity and commutativity may fail depending on the operation, but bilinearity is common to every "product". This can be regarded as the essential condition for being called a product.
Note: a group operation does not have bilinearity (it is not necessarily defined on a linear space). Bilinearity is a property specific to "products" on linear spaces.