Rings and Fields - Introductory Abstract Algebra (Undergraduate Advanced)

This course introduces rings and fields—two central structures in abstract algebra. You’ll learn how rings extend the idea of groups with two operations, and how fields bring in the familiar arithmetic of fractions and equations. We explore key examples like integers, polynomials, and modular systems. Simple, structured, and made for first-time learners building up their algebra toolkit.

Payment required for enrolment
Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[University] Introduction to Abstract Algebra
[University] Introduction to Abstract Algebra
Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Course Chapters

1. Introduction

Definition of a ring, ring axioms, properties, examples.

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2. Some Special Rings

Commutative rings, Boolean rings, integral domains - properties and examples.

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3. Subrings

Definition, subring test, and illustrative examples; two-sided ideals.

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4. Ring Homomorphisms

Definition and illustration of ring homomorphisms.

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5. Fields

Division rings, fields - definition, properties and illustrative examples.

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