UM 205: Introduction to algebraic structures
Credits: 3:1
- Set theory: equivalence classes, partitions, posets, axiom of choice/Zorn’s lemma, countable and uncountable sets.
- Combinatorics: induction, pigeonhole principle, inclusion-exclusion, Möbius inversion formula, recurrence relations.
- Number theory: Divisibility and Euclids algorithm, Pythagorean triples, solving cubics, Infinitude of primes, arithmetic functions, Fun- damental theorem of arithmetic, Congruences, Fermat’s little theorem and Euler’s theorem, ring of integers modulo n, factorisation of poly- nomials, algebraic and transcendental numbers.
- Graph theory: Basic definitions, trees, Eulerian tours, matchings, matrices associated to graphs.
- Algebra: groups, permutations, group actions, Cayley’s theorem, di- hedral groups, introduction to rings and fields.
Suggested books and references:
- L. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer-Verlag.
- M. A. Armstrong, Groups and Symmetry, Springer-Verlag.
- Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, World Scientific.
- D. M. Burton., Elementary Number Theory, McGraw Hill.
- Niven, Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edition, Wiley Student Editions.
- Fraleigh, G., A First Course in Abstract Algebra, 7th edition, Pearson.