Fields and Geometry III
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This first part of this course generalizes the real numbers to a mathematical structure called a field. Finite fields have many applications, particularly in Information Security where the understanding of finite fields is fundamental to many codes and cryptosystems. Properties and constructions of fields will be investigated in detail. The second part of the course considers projective geometries. Projective geometry is one of the important modern geometries introduced in the 19th century. Projective geometry is more general than our usual Euclidean geometry, and it has useful applications in Information Security, Statistics, Computer Graphics and Computer Vision. The focus of this course will be primarily on projective planes.
To provide an introduction to the areas of Fields and
Projective Geometry with particular emphasis on the links between the
two areas. At the end of this course students should:
have a knowledge of the structure of finite fields and be able to
perform basic calculations in finite fields.
understand the ideas in projective geometry, and how projective
geometry relates to Euclidean geometry.
have enough tools to study objects and transformations in
projective planes corresponding to fields.
Topics covered are: (I) Fields: fields, polynomials rings, extensions of fields; automorphisms of fields, the structure of a finite field. (II) Projective Geometry: projective planes, homogeneous coordinates, field planes, collineations of projective planes, conics in field planes, projective geometry of general dimension.
36 hours of lectures and tutorials
Ongoing assessment 30%, exam 70%.
Prerequisite is MATHS 1007A/B Mathematics I (Pass
Div I) or both MATHS 1007A/B Mathematics I (Pass Div II) and MATHS
2004 Mathematics IIM (Pass Div I). It will be an advantage to have
done PURE MTH 2002 Algebra II, although the necessary material is
revised at the start of the course.
This course complements the first semester
course PURE MTH 3007 Groups and Rings III. It also contains
concepts that are useful for the course PURE MTH 3006 Coding and
This course is one of the core Pure Mathematics
courses, and provides a strong foundation for further study in the
areas of Algebra and Projective Geometry. Finite fields have many
applications, and an understanding of their structure is essential
to students who want to further their knowledge of codes and