This free course 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. 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. Section 3 develops results concerning the divisibility of polynomials.
Course learning outcomes
After studying this course, you should be able to:
recall and be able to use the axioms that define a ring, and know the basic properties of rings arising from these axioms
know how to add and multiply polynomials over arbitrary fields, and be able to use this to define polynomial rings
understand the statement and proof of the Division Algorithm for polynomials, and be able to apply polynomial long division in the ring Q[x]
understand the meaning of the highest common factor of two polynomials, the proof of existence of the hcf, the meaning of ‘coprime’ in the context of polynomials over fields, and be able to apply the Euclidean Algorithm to compute the hcf of two polynomials f and g in Q[x], and find polynomials a, b such that hcf(f, g) = af + bg
understand the meaning of the least common multiple of two polynomials, the proof of its uniqueness, and be able to compute lcms in the polynomial ring Q[x].
You can start this course right now without signing-up. Click on any of the course content sections below to start at any point in this course.
If you want to be able to track your progress, earn a free Statement of Participation, and access all course quizzes and activities, sign-up.
Creative commons: The Open University is proud to release this free course under a Creative Commons licence.
However, any third-party materials featured within it are used with permission and are not ours to give away. These
materials are not subject to the Creative Commons licence. See terms and conditions377 and our FAQs378.
Full copyright details can be found in the Acknowledgements section of each week.
For further information, take a look at our frequently asked questions which may give you the support you need.
Making the decision to study can be a big step, which is why you'll want a trusted University.
The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now.
Take a look at all Open University courses.
If you are new to University-level study,
we offer two introductory routes to our qualifications. You could either choose to start with an
Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on
Where to take your learning next for more information.