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Content

The first part of the course continues the treatment of Groups started in ABSTRACT ALGEBRA I. Some important subgroups are defined, and the important concept of a group acting on a set is introduced. The power of group actions is demonstrated by using the technique to prove several key results about finite groups. The investigation of finite groups is concluded with the famous Sylow Theorems. The construction of the (finite) direct product should be familiar to any mathematician, and so the course proceeds to do this. Abelian groups are discussed briefly; a statement of the Decomposition Theorem for finite groups is given. The section on Group Theory is concluded with a discussion of subgroup series – an important technique in determining the structure of a group. The Jordan-Holder Theorem is proved, and an important class of groups - the solvable groups are introduced. The course then shifts focus to one of the most important examples of a Euclidean ring – the polynomial ring over a field. (Euclidean rings were introduced in ABSTRACT ALGEBRA I.) The fundamental results that transfer from Euclidean rings are restated in context, and the idea of irreducibility is introduced. The course then specialises to the rational field, and several key results concerning polynomials over the rationals are proved. The course naturally progresses to investigate the existence of roots of polynomials over their base field. The extremely important construction of the algebraic extension containing the root of a polynomial is done in detail, with several interesting and motivating examples. The course continues to prove the existence of a splitting field and concludes with a statement of the Fundamental Theorem of Algebra. Straightedge and compass constructions will be presented as an application if time permits. Group theory. Conjugates and commutators. Group actions. Finite groups. The Sylow theorems. Direct products. Composition series. The Jordan-Holder Theorem. Solvable groups. Polynomial rings over a field. Polynomials over the rationals. Field of fractions. Algebraic extensions. Splitting fields. Straightedge and compass constructions.

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Conditions of Participation

Math 2272 and Math 2273 are prerequisites.

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Teaching Methods

Myelearning will be used to foster communication between students and lecturer, as well as to post relevant course materials such as lecture notes, problem sheets, and worked problems. Lectures/tutorials will be given 3 times per week on Blackboard Collaborate.

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Learning outcomes / Competences

Upon successful completion of the course, students will be able to Construct new groups from existing groups Manipulate groups of permutations Define p-groups and appreciate their fundamental importance Define composition series and prove the Jordan-Holder theorem Define solvable groups in terms of solvable series Manipulate rings and ideals Give examples of rings that occur naturally in mathematics State the formal definition of a polynomial Manipulate and factorise polynomials over the rational field Construct and investigate algebraic extensions of fields Prove the existence of splitting fields Formulate non-trivial problems in an algebraic context

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Forms of examination

Final exam (50%) 2 coursework exams (40%) 7 assignments (10%)

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Conditions for allocation of credit points

Passing grades will be given for an overall score of at least 50%

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Other information

This is a core course for the B.Sc. Math (Special) degree and it is an elective for any other degree.