# Ch 19: Algebra: Number Theory & Abstract Algebra

### About This Chapter

## Algebra: Number Theory & Abstract Algebra - Chapter Summary

This chapter can help you review abstract algebraic theorems, operations and functions. The expert instructors of this series of lesson videos will guide you through example problems involving divisibility, the Division Algorithm, Fermat's Last Theorem and abelian groups. This chapter also covers topics of standard sets in abstract algebra and group homomorphisms. After watching the lessons in this chapter you should have a mastery over material on:

- Fundamental ideals of number theory
- Euclidean algorithm and the diophantine equations
- Binary and non-binary structures and operations
- Algebraic subgroups
- Isomorphism and homomorphism
- Finite abelian groups
- Rings

To effectively master the material of this chapter, ask our instructors any questions you may come up with about the lessons. Increase your understanding of the material in these lesson videos by completing quizzes and printing out the lesson worksheets to use as a study aid. With the results of your lesson quizzes, determine your areas of weakness, and then fortify these areas by reviewing key points in the lesson videos via video tags on the timeline. Reading over the lesson transcripts may also prove to be beneficial as they present the material in an alternative written format.

### 1. Number Theory: Divisibility & Division Algorithm

In this lesson, we'll define the division algorithm and divisibility. We'll see how these two concepts are related and use examples to explore some different divisibility rules to add to your math toolbox.

### 2. Euclidean Algorithm & Diophantine Equation: Examples & Solutions

Finding the greatest common divisor of two numbers is nicely accomplished using the Euclidean algorithm. In this lesson, we use this algorithm to explore solutions of Diophantine equations.

### 3. Fermat's Last Theorem: Definition & Example

Considered a mathematical mystery for hundreds of years, watch this video lesson to learn what Fermat's Last Theorem tells us and why mathematicians struggled for years to prove it true.

### 4. Binary Operation & Binary Structure: Standard Sets in Abstract Algebra

In mathematics, we like to combine things together in different ways to try to figure out what they make. We use binary operations to combine things together, and understanding such operations is vital in the study of abstract algebra.

### 5. Binary and Non-Binary Operations

Binary and non-binary are two terms that might really scare you, much like an operation may scare you. But don't worry, this lesson is super fun and super easy. You'll learn the definitions of these terms and see examples of each.

### 6. Types of Subgroups in Abstract Algebra

When looking at groups in abstract algebra, we can often find smaller groups nested within other groups. These smaller groups are called subgroups, and there are some special types of subgroups that are important in abstract algebra.

### 7. Finitely Generated Abelian Groups: Classification & Examples

In this lesson, you'll explore the definition of finitely generated abelian groups and how they can be classified. You'll also be introduced to some examples and their classification.

### 8. Group Homomorphisms: Definitions & Sample Calculations

We're going to talk about a special type of function on a group called a homomorphism. This special function not only maps elements from one group to another, it actually preserves some properties of the group as we map from one to another.

### 9. Rings: Binary Structures & Ring Homomorphism

In this lesson, basic definitions are presented for algebraic rings and ring homomorphisms. These form the foundations for a broader discourse in abstract algebra. Rings are used in more complex algebraic structures and have many applications in computer science, physics, and engineering.

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### Other Chapters

Other chapters within the GRE Math: Study Guide & Test Prep course

- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards