Mathematics
MA491: Field Theory (5 ECTS)
This is an introduction to the theory of Field Extensions, their Galois groups and the application of finite fields to constructing BCH codes.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 100 hours (24 Lecture hours, 12 Tutorial hours, 64 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- State the definition of a field and finite extensions of a field.
- Compute the degree of such an extension.
- Define the notion of algebraic number, transcendental number, the algebraic closure of a field.
- State the 3 famous problems of ancient Greek geometry, Ruler and Compass constructions, and how Field Theory contributes to answering these problems and also the construction of regular n-gons via roots of cyclotomic polynomials.
- Define the notion of automorphisms of an extension field relative to the field of rational numbers.
- Be able to construct the splitting field of an irreducible polynomial over Q and the corresponding Galois group of automorphisms in the case of small degree.
- Know why there is no general formula for "solving the quintic by radicals" and what that expression means.
- Construct finite fields of small order, properties of finite fields, the Frobenius automorphism. The formula of Gauss for the number of monic irreducible polynomials of degree n over a given finite field.
- Use finite fields in constructing BCH codes of a designated distance d over such a field.
Indicative Content
This course together with Ring Theory form the basis of modern Abstract Algebra, and as such is part of the indispensible knowledge of an honours graduate in Mathematics, regardless of whether or not further mathematical studies are pursued. In particular Field Theory and Galois Theory provide answers for questions in mathematics which originate 2,000 years earlier in the case of Greek Geometry, and some 450 years ago relating to formulas for the roots of polynomials of degree 5 and higher. The notion of Galois correspondence occurs in other branches of modern and abstract algebra. More recently, with the wide applicability of Coding Theory and the search for optimal codes, the use of finite (Galois) fields has assumed a critical importance.
Module Resources
- Rings and Fields, Graham Ellis (OUP)
- A First Course in Abstract Algebra (7th ed), John B. Fraleigh (Addison Wesley)
- Galois Theory (2nd ed.), David A. Cox (John Wiley)
- Galois Theory (3rd ed.) Ian Stewart (Chapman & Hall)
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