MAT1110: Algebraic Structures

Lectures: 9am Wednesdays and 12pm Fridays in LT6 for 12 weeks, starting Wednesday 22 January 1997.

Lecturer: Dr Robin Chapman, room 811, email: rjc@maths,

   web page: http://www.maths.ex.ac.uk/˜ rjc/rjc.html .

Brief syllabus: Functions and Relations: More on functions: composition of functions, identity and inverse functions. Relations, equivalence relations, equivalence classes. Numbers: More on the natural numbers and integers. Divisibility and primes. Euclid’s proof of the infinitude of primes. Modular arithmetic and congruences. The division algorithm. Greatest common divisors and the Euclidean algorithm. The unique factorization theorem. Solution of linear congruences. The Chinese Remainder Theorem and the solution of simultaneous congruences. Permutations: The notion of a permutation. Arithmetic of permutations. Cycle notation. Order of permutations. Representation as products of transpositions. The sign of a permutation. Groups: The notion of a group. Examples: Z, Z_{n}, S_{n} and A_{n}. Cyclic groups. Subgroups. Orders of elements and of subgroups. (Time permitting) Lagrange’s Theorem. Polynomials: Arithmetic of polynomials. Degree. Division algorithm and remainder theorem. Euclidean algorithm, and factorization of polynomials.

Recommended Book: Norman L. Biggs, Discrete Mathematics, (2nd edition) Oxford University Press, 1989.

There are many other books covering the course material, and other topics besides. You may like to look in both the main library and the Roborough library, or ask your tutors for their recommendations.

Course Work: Fortnightly example sheets, to be handed in to and discussed with tutors.

Examination: There will be a 50-minute test at 11am on 13th March 1997, counting as 20% of the final assesment. The remaining 80% will be based on a two-hour examination, with five questions. The best three answers count.

Office Hours: See my door (811) or make an appointment by sending me a note or an email.