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Algebra has a long history of important applications throughout mathematics, science and engineering, and is also studied for its intrinsic beauty. In this subject we study the algebraic laws satisfied by familiar objects such as integers, polynomials and matrices. This abstraction simplifies and unifies our understanding of these structures and enables us to apply our results to interesting new cases. Students will gain further experience with abstract algebraic concepts and methods. General structural results are proved and algorithms developed to determine the invariants they describe.
Intended learning outcomes
On completion of this subject, students should
Have an understanding of:
- rings, factorization in rings, principal ideal domains, Euclidean domains;
- modules, free modules, structure theorem for finitely generated modules over a principal ideal domain;
- fields, field extensions, finite fields, Galois extensions; splitting fields and the Galois correspondence.
Be able to:
- apply the Euclidean algorithm in a general context, including polynomials;
- prove results about rings, modules and fields;
- describe the Galois correspondence for the splitting field of a polynomial.
- calculate the Jordan Normal form of a matrix;
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 7 September 2023