Rings and polynomials

Introduction

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This free OpenLearn course, Rings and polynomials, is an extract from the Open University course M303 Further pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level course that introduces important topics in the theory of pure mathematics including: number theory; the algebraic theory of rings and fields; and metric spaces. Students studying M303 develop their understanding of group theory and real analysis and see how some of the ideas are applied to cryptography and fractals.

Rings and polynomials consists of material from M303 Book C, Chapter 11 and has three sections in total. You should set aside about five hours to study each of the sections. Note that while Section 1 is quite long it contains relatively few proofs and that Section 2 is shorter than the other two sections. The whole extract should take about 16 hours to study. The extract is a small part (around 4%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

This extract contains an introduction to rings and polynomials. We see that polynomial rings have many properties in common with the integers; for example, we can define a division algorithm, and this enables us to develop the analogue of the highest common factor for two polynomials. It is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of pure mathematics, such as could be gained from studying the Open University course M208 Pure mathematics. A few techniques and definitions are present in the extract without explanation.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M303 video material (although please note that the PDF may contain references to other parts of M303). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 explores the abstract definitions of a ring and a field.

Sections 2 and 3 define polynomial rings where the coefficients of the polynomials are elements from a given field. These polynomials have many properties in common with the integers.

Section 3 develops results concerning the divisibility of polynomials.