Leader: Associate Professor Hans Lausch
Clayton Second semester 2006 (Day)
Synopsis: Rings, fields, algebraic integers, finite fields, splitting fields of polynomials and fields of fractions. Classical problems of ruler and compass (eg. can an angle be trisected?). Coding, cryptography, and geometric constructions. Gaussian integers, Hamilton's quaternions, Chinese Remainder Theorem. Euclidean Algorithm in further fields
Objectives: 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; 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 , 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 or Gaussian integers; constructing larger fields from smaller fields (field extensions); applying ring and field theory to coding, cryptography and geometric constructions.
Assessment: Examination (3 hours): 70% + Assignments and tests: 30%
Contact Hours: Three 1-hour lectures and an average of one 1-hour support class per week