Free course

Rings and polynomials

Rings and polynomials Copyright free Icon

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].

First Published: 09/03/2018

Updated: 23/05/2018

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.

Course content

Skip Rate and Review

Take your learning further

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.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371