## MATH342 Modern Algebra

First Semester |

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!)

### Paper details

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!

### Potential students

This paper should be of interest to anyone who wishes to see how algebraic properties arising in different branches of pure mathematics can be described using the unifying concepts of a group and a ring.

Students who wish to pursue their interests in algebra should take this course as a foundation to more advanced papers in the theory of groups, Galois Theory, rings, modules and algebras.

### Prerequisites

MATH 202

### Main topics

- A review of functions; equivalence relations; modular arithmetic.
- Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange’s theorem; Application - Public key cryptography.
- Rings; subrings; integral domains; matrix rings; polynomial rings; homomorphism and isomorphism; ideals; quotient rings; fields; vector spaces; Application - Error correcting codes.

### Required text

No required text - Comprehensive Course notes will be provided.

### Lecturer

Professor Mike Hendy, Room 517

### Lectures

Monday at 10:00, Wednesday at 11:00 and alternate Fridays at 11:00. (Location to be announced.)

### Tutorial

Thursday at 9.00am in room MA241.

### Internal Assessment

There will be 5 exercises making up 50% of the internal assessment. You will be encouraged to use the mathematical formatting language LaTeX for your assignments.

The remaining 50% of internal assessment will come from two 45 minute written tests. These are scheduled for Friday April 7 and Friday May 12 (11.00 - 11.45am)

### Exam format

The final examination is 3 hours long.

### Final mark

If:

- E is the Exam mark
- A is the Assignments mark
- T is the Tests mark

all expressed out of 100, then your final mark F in this paper will be calculated according to this formula:

**F = max(0.85E + 0.075A + 0.075T, 0.70E + 0.15A + 0.15T)**

Thus your internal assessment contributes either 15% or 30% towards your final mark.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### Joseph Louise Lagrange...

..., 1736-1813, was born in Italy although his family was originally from France. In 1766 he went to St. Petersburg, Russia, at the following request of Tsar Frederick the Great: “The greatest king in Europe wishes to have at his court the greatest mathematician in Europe”. He stayed there until Frederick’s death in 1788, moving to Paris at the invitation of King Louis XIV. Lagrange’s Theorem about the size of subgroups of a finite group is one of the most important results in algebra.

### Quaternions

In 1833, the Irish mathematician

**William Rowan Hamilton**gave one of the first algebraic descriptions of the set of complex numbers. Of course, each complex number can be described as a sum

*a*+

*ib*where

*a*and

*b*are real numbers and

*i*is the “imaginary” number with the property that its square is -1. Also such a number can be thought of as the point on the two-dimensional

*x-y*plane with

*a*as its

*x*-coordinate and

*b*as its

*y*-coordinate. Now, Hamilton tried for ten years to find a similar way of algebraically describing three-dimensional space. On October 6, 1843, while out walking in Dublin, he finally realized that there was no algebraic three-dimensional analogue but that there was a four-dimensional one. He formed a new set of numbers called the quaternions in which there are four key ingredient numbers, namely 1, $i$, $j$, and $k$, satisfying the following multipicative rules:

$$i^2=j^2=k^2=-1\\ij = k, jk = i, ki = j,\\ji = -k, kj = -i, ik = -j$$ Hamilton was so pleased with his discovery that he stopped on his walk to carve these equations with a knife into the sandstone of Brougham Bridge (see Irish stamp above). The quaternions give us important examples of both a group and a division ring.