# Course Name

Modern Algebra

Session: VDNS3121

18 Credit Points

# Prerequisites & Language Level

Taught In English

• There is no language prerequisite for courses at this language level.

# Overview

Introduces the modern algebraic concepts of a group and a ring. These concepts occur throughout modern mathematics and this paper looks at their properties and some applications.

Modern algebra is studied all over the world, perhaps not surprising in view of its international beginnings in the late 1700s work of the Swiss mathematician Leonhard Euler, the French mathematician Joseph Louis Lagrange, and the German mathematician Carl Friedrich Gauss. Their work led to the introduction in the 1800s of the unifying abstract algebraic concepts of a group and a ring, the first of these pioneered by the British algebraist Arthur Cayley, the second due to Richard Dedekind, also German. These two notions of a group (a set with a standard operation, usually called multiplication) and a ring (a set with two operations, usually called addition and multiplication) occur throughout modern mathematics in both its pure and applied branches and, even after more than 100 years since their introduction, most of today's research in modern algebra involves the study of either groups or rings (or both!)

The learning aims of the paper are principally to develop the notions of a group and ring, to see how these arise in a variety of mathematical settings, and to establish their fundamental properties. Since this is a Pure Mathematics paper which will provide the basis for further study in abstract algebra, concepts will be introduced and developed rigorously. We will be doing a lot of proofs!

Course Structure
Main topics:
• A review of functions; equivalence relations; modular arithmetic
• Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange's theorem; group actions
• Rings; subrings; integral domains; matrix rings; polynomial rings; homomorphism and isomorphism; ideals; quotient rings; The Chinese Remainder theorem

Learning Outcomes: Demonstrate the ability to use mathematical reasoning by writing proofs in the context of groups and rings.

*Course content subject to change