MTH3150 - Algebra and number theory II
6 points, SCA Band 0 (NATIONAL PRIORITY), 0.125 EFTSL
Undergraduate Faculty of Science
Leader(s): Dr Ian Wanless
Clayton Second semester 2009 (Day)
Rings, fields, ideals, algebraic extension fields. Coding theory and cryptographic applications of finite fields. Gaussian integers, Hamilton's quaternions, Chinese Remainder Theorem. Euclidean Algorithm in further fields.
At the completion of this unit, students will be able to demonstrate understanding of advanced concepts, algorithms and results in number theory; the use of Gaussian integers to find the primes expressible as a sum of squares, Diophantine equations, primitive roots; the quaternions, the best known skew field; many of the links between algebra and number theory; the most commonly occurring rings and fields: integers, integers modulo n, rationals, reals and complex numbers, more general structures such as algebraic number fields, algebraic integers and finite fields; and will have developed skills in the use of the Chinese Remainder Theorem to represent integers by their remainders; performing calculations in the algebra of polynomials; the use of the Euclidean algorithm in structures other than integers; constructing larger fields from smaller fields (field extensions); applying field theory to coding and cryptography.
Examination (3 hours): 70%
Assignments and tests: 30%
Three 1-hour lectures and an average of one 1-hour support class per week