MATH321-23S1 (C) Semester One 2023

# Rings and Fields

15 points

Details:
 Start Date: Monday, 20 February 2023 End Date: Sunday, 25 June 2023
Withdrawal Dates
Last Day to withdraw from this course:
• Without financial penalty (full fee refund): Sunday, 5 March 2023
• Without academic penalty (including no fee refund): Sunday, 14 May 2023

## Description

An introduction to fields and rings, including applications to coding theory and the impossibility of constructions such as ‘squaring the circle’.

This course formally introduces rings and fields, which have been encountered at 100- and 200-level in special situations, and investigates their algebraic structure. It gives a deeper understanding of these algebraic concepts and thus provides a thorough grounding in the algebraic theory which underpins modern applications like cryptography, error-correcting codes, number theory or finite mathematics. If you are interested in any of these subjects or if you want to see how algebraic theory can be applied to solve certain geometric construction problems or prove their impossibility, then this is the course to take.

The topics covered by this course are:

• fundamentals of ring theory: subrings, ideals, factor rings, ring homomorphisms;
• special rings: integral domains and polynomial rings and factorizations of elements therein;
• fundamentals of field theory: field extensions, constructions of fields, in particular finite fields, and their uses, like the impossibility of certain geometric constructions such as trisecting the angle.

## Learning Outcomes

• Students successfully completing this course should:

• understand a range of basic algebraic concepts.
• have developed a high level of competence at core algebraic skills.
• be able to confidently apply algebraic concepts in practical settings.
• be able to present clear and logical mathematical arguments.

## Prerequisites

One of MATH203, MATH220, MATH240, or
EMTH211, and a further 15 points from MATH201-294.

MATH439, MATH311

## Timetable 2023

Students must attend one activity from each section.

Activity Day Time Location Weeks Lecture A 01 Wednesday 10:00 - 11:00 Jack Erskine 446 20 Feb - 2 Apr 24 Apr - 4 Jun Lecture B 01 Tuesday 10:00 - 11:00 Beatrice Tinsley 112 (21/2) Jack Erskine 315 (28/2-28/3, 25/4-30/5) 20 Feb - 2 Apr 24 Apr - 4 Jun Tutorial A 01 Thursday 15:00 - 16:00 Jack Erskine 443 20 Feb - 2 Apr 24 Apr - 4 Jun

## Examination and Formal Tests

Activity Day Time Location Weeks Test A 01 Wednesday 18:30 - 19:30 Ernest Rutherford 465 27 Mar - 2 Apr

## Assessment

Assessment Due Date Percentage
Assignments 25%
Test 25%
Final Examination 50%

To obtain a pass (C- or better), you must obtain at least 40% in the exam.

## Textbooks / Resources

Joseph Gallian; Contemporary Abstract Algebra ; 10th (or earlier edition); Taylor & Francis Ltd, 2021.

## Indicative Fees

Domestic fee \$824.00

International fee \$4,750.00

* All fees are inclusive of NZ GST or any equivalent overseas tax, and do not include any programme level discount or additional course-related expenses.

For further information see Mathematics and Statistics .

## All MATH321 Occurrences

• MATH321-23S1 (C) Semester One 2023