Fields and Geometry III

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Description

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.

Objective

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.

Content

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.

Delivery

36 hours of lectures and tutorials

Assessment

Ongoing assessment 30%, exam 70%.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)

Linkage past

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.

Linkage present

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 Cryptology III

Linkage future

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 cryptosystems.

Restrictions

None.

Recommended text

None.