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Rings and polynomials

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# Conclusion

Following completion of this free OpenLearn course, Rings and polynomials, as well as being able to understand the terms and definitions, and use the results introduced, you should also find that your skills and confidence in reading, understanding and writing mathematical arguments are improving.

You should now 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].

This free OpenLearn course 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)] .

If you feel you are ready to move on in your study of pure mathematics but don’t have time to study a full Open University course at this time, you might like to study one of the free OpenLearn courses Number theory, Group theory and Metric spaces and continuity. They are further adapted extracts from the Open University course M303 Further pure mathematics.

If you are interested in applied mathematics, you might like to study the free OpenLearn courses Introduction to the calculus of variations, Linear programming – the basic ideas and Kinematics of fluids.

If you are interested in statistics, you might like to study the free OpenLearn courses Modelling events in time, Univariate continuous distribution theory and Point estimation.