This course is a continuation of Abstract Algebra I: the student will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms. The student will also take a look at ring factorization, general lattices, and vector spaces. Later this course presents more advanced topics, such as Galois theory - one of the most important theories in algebra, but one that requires a thorough understanding of much of the content we will study beforehand. Upon successful completion of this course, students will be able to: Compute the sizes of finite groups when certain properties are known about those groups; Identify and manipulate solvable and nilpotent groups; Determine whether a polynomial ring is divisible or not and divide the polynomial (if it is divisible); Determine the basis of a vector space, change bases, and manipulate linear transformations; Define and use the Fundamental Theorem of Invertible Matrices; Use Galois theory to find general solutions of a polynomial over a field. (Mathematics 232)

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

Access also available here: http://abstract.ups.edu/contact.html

Table of Contents
Preliminaries
The Integers
Groups
Cyclic Groups
Permutation Groups
Cosets and Lagrange's Theorem
Introduction to Cryptography
Algebraic Coding Theory
Isomorphisms
Normal Subgroups and Factor Groups
Homomorphisms
Matrix Groups and Symmetry
The Structure of Groups
Group Actions
The Sylow Theorems
Rings
Polynomials
Integral Domains
Lattices and Boolean Algebras
Vector Spaces
Fields
Finite Fields
Galois Theory

Intermediate Algebra is the second part of a two-part course in Algebra. Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study of applications found in most disciplines. Used as a standalone textbook, it offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. While CAS independent, a standard scientific calculator will be required and further research using technology is encouraged.

Intermediate Algebra clearly lays out the steps required to build the skills needed to solve a variety of equations and interpret the results. With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems. In addition to embedded video examples and other online learning resources, the importance of practice with pencil and paper is stressed. This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. In addition, Intermediate Algebra was written from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success.

The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in all disciplines. After completing a course sequence based on Elementary and Intermediate Algebra, students will be on firm footing for success in higher-level studies at the college level.

" The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc."

This course discusses how to use algebra for a variety of everyday tasks, such as calculate change without specifying how much money is to be spent on a purchase, analyzing relationships by graphing, and describing real-world situations in business, accounting, and science.

This course is oriented toward US high school students. Its structure and materials are aligned to the US Common Core Standards. Foci include: graphing, equations and inequalities.

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Algebra and Trigonometry provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra and trigonometry course. The modular approach and the richness of content ensures that the book meets the needs of a variety of courses. Algebra and Trigonometry offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.

Table of Contents
1 Prerequisites
2 Equations and Inequalities
3 Functions
4 Linear Functions
5 Polynomial and Rational Functions
6 Exponential and Logarithmic Functions
7 The Unit Circle: Sine and Cosine Functions
8 Periodic Functions
9 Trigonometric Identities and Equations
10 Further Applications of Trigonometry
11 Systems of Equations and Inequalities
12 Analytic Geometry
13 Sequences, Probability, and Counting Theory

Access also available here: https://openstax.org/details/books/algebra-and-trigonometry

Algebra and Trigonometry provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra and trigonometry course. The modular approach and the richness of content ensures that the book meets the needs of a variety of courses. Algebra and Trigonometry offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.

" This is an introductory course in algebraic combinatorics. No prior knowledge of combinatorics is expected, but assumes a familiarity with linear algebra and finite groups. Topics were chosen to show the beauty and power of techniques in algebraic combinatorics. Rigorous mathematical proofs are expected."

" This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry."

In this second term of Algebraic Topology, the topics covered include fibrations, homotopy groups, the Hurewicz theorem, vector bundles, characteristic classes, cobordism, and possible further topics at the discretion of the instructor.

Table of Contents:

Chapter 1: Algebra Review
Introduction to Algebra Review
Section 1.1 Functions
Section 1.2 Operations on Functions
Section 1.3 Linear Functions
Section 1.4 Exponents
Section 1.5 Quadratics
Section 1.6 Polynomials and Rational Functions
Section 1.7 Exponential Functions
Section 1.8: Logarithmic Functions
Chapter 1 Review Exercises
Chapter 1 Review Problems

Chapter 2: Limits and The Derivative
Introduction to Limits
Section 2.1: Limits and Continuity
Introduction to the Derivative
Section 2.2: The Derivative
Section 2.3: The Power and Sum Rules for Derivatives
Section 2.4: Product and Quotient Rules
Section 2.5: Chain Rule
Section 2.6: Second Derivative and Concavity
Chapter 2 Review Exercises: Limits
Chapter 2 Review Exercises: The Derivative
Chapter 2 Review Problems

Chapter 3: Applications of the Derivative
Introduction to Applications of the Derivative
Section 3.1: Optimization
Section 3.2: Curve Sketching
Section 3.3: Applied Optimization
Section 3.4: Other Applications
Section 3.5: Implicit Differentiation and Related Rates
Chapter 3 Review Exercises
Chapter 3 Review Problems
Chapter 3 Solutions to Review Problems

Chapter 4: The Integral
Introduction to the Integral
Section 4.1: The Definite Integral
Section 4.2: The Fundamental Theorem and Antidifferentiation
Section 4.3: Antiderivatives of Formulas
Section 4.4: Substitution
Section 4.5: Average Value and the Net Change Theorem
Section 4.6: Applications to Business
Chapter 4 Review Exercises
Chapter 4 Review Problems

This textbook was created through Connecting the Pipeline: Libraries, OER, and Dual Enrollment from Secondary to Postsecondary, a $1.3 million project funded by LOUIS: The Louisiana Library Network and the Institute of Library and Museum Services. This project supports the extension of access to high-quality post-secondary opportunities to high school students across Louisiana and beyond by creating materials that can be adopted for dual enrollment environments. Dual enrollment is the opportunity for a student to be enrolled in high school and college at the same time.

The cohort-developed OER course materials are released under a license that permits their free use, reuse, modification and sharing with others. This includes a corresponding course available in Moodle and Canvas that can be imported to other platforms.

Laszlo Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction the original lecture notes for the course "Applied Geometric Algebra" taught by Professor Tisza in the Spring of 1976. Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems. The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, '77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publication-ready PDFs. You can read about his motivation for wanting to see these notes published in his Preface below. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.

Arithmetic | Algebra provides a customized open-source textbook for the math developmental students at New York City College of Technology. The book consists of short chapters, addressing essential concepts necessary to successfully proceed to credit-level math courses. Each chapter provides several solved examples and one unsolved “Exit Problem”. Each chapter is also supplemented by its own WeBWork online homework assignment. The book can be used in conjunction with WeBWork for homework (online) or with the Arithmetic | Algebra Homework handbook (traditional). The content in the book, WeBWork and the homework handbook are also aligned to prepare students for the CUNY Elementary Algebra Final Exam (CEAFE).

Arithmetic | Algebra Homework book is a static version of the WeBWork online homework assignments that accompany the textbook Arithmetic | Algebra for the developmental math courses MAT 0630 and MAT 0650 at New York City College of Technology, CUNY.

Chapter 1. Concepts
Chapter 2. Linear Equations
Chapter 3. Graphing
Chapter 4. Linear Systems
Chapter 5. Polynomial Equations
Chapter 6. Factoring Polynomials
Chapter 7. Rational Expressions
Chapter 8. Radicals
Chapter 9. Quadratic Equations

This course is also intended to provide the student with a strong foundation for intermediate algebra and beyond. Upon successful completion of this course, you will be able to: simplify and solve linear equations and expressions including problems with absolute values and applications; solve linear inequalities; find equations of lines; and solve application problems; add, subtract, multiply, and divide various types of polynomials; factor polynomials, and simplify square roots; evaluate, simplify, multiply, divide, add, and subtract rational expressions, and solve basic applications of rational expressions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 001)

In this beginning algebra course, you'll learn about
fundamental operations on real numbers
exponents
solving linear equations and inequalities
applications
functions
graphing linear equations
slope
systems of linear equations