Module MATH3511: Galois Theory, Groups and Geometry III

Department: Mathematical Sciences

MATH3511: Galois Theory, Groups and Geometry III

Prerequisites

• Algebra II (MATH2781 or MATH2581) AND Complex Analysis II (MATH2791 or MATH2011)

Corequisites

• None

Excluded Combination of Modules

• Galois Theory III (MATH3041) AND Geometry III (MATH3201)

Aims

• To introduce further concepts in abstract algebra, particularly the usage of symmetry groups and their applications in Galois Theory and Geometry.

Content

• Groups and Galois Theory:
• Fields and polynomials, algebraic numbers and minimal polynomials.
• Algebraic extensions, field automorphisms, normal and separable extensions, Galois groups and the Fundamental Theorem of Galois Theory.
• Finite fields: existence, uniqueness, constructions and Galois groups of extensions.
• Galois groups of polynomials. Explicit solutions of cubic and quartic equations. Criterion for solvability by radicals.
• Further applications: Cyclotomic fields, ruler-and-compass constructions.
• Groups and Geometry:
• Euclidean Geometry: Isometry group, generators, fixed points.
• Spherical Geometry: Distance, isometries, area of triangles.
• Projective Geometry: Projective line, projective plane.
• Hyperbolic geometry: Conformal models, Poincaré disc and upper half-plane models. Elementary hyperbolic geometry, sine and cosine rules, area of a triangle, isometries. Projective models, Klein model and hyperboloid model.
• Further applications: Hyperbolic surfaces.

Learning Outcomes

• By the end of the module students will:
• Be able to solve seen and unseen problems on the given topics.
• Have an awareness of the abstract concepts of theoretical mathematics in the fields of Algebra and Geometry.
• Have a knowledge and understanding of Galois Theory and Geometry.

• In addition, students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: abstract reasoning.

• Students will have basic mathematical skills in the following areas: abstract reasoning, problem solving.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• Summative assignments test achievement of learning outcomes and provide feedback to students about their mastery of the topics.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four assignments to be submitted

■ Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.