Text: A. Papantonopolou, Algebra Pure and Applied
| (a)
The Integers Division algorithm. Greatest common divisors. Euclidean algorithm. Fundamental Theorem of Arithmetic. Congruence Arithmetic. Applications to RSA cryptography. | [5] |
| (b) Permutations Cycle decomposition. Order of a permutation. Even and odd permutations. | [3] |
| (c) Groups Definition and examples. Subgroups, cosets and Lagrange’s Theorem. (Application to Key Exchange Protocol). Normal subgroups and quotient groups. Homomorphisms. The Isomorphism Theorems with illustrations. | [11] |
| (d) Group Actions Examples of group actions. Orbits and stabilizers. Burnside’s Lemma. Applications to Cayley’s Theorem, the class equation, centers of finite p-groups, Cauchy’s Theorem. Simplicity of A5. Applications to Polya counting theory. | [9] |
| (e) Rings Definition and Examples. Polynomial rings. Subrings and ideals. Quotient rings, homomorphisms. Isomorphism Theorems for rings. Integral domains and fields. Field of fractions of a domain. Division algorithm for F [ x ], F a field. Roots of polynomials. The Remainder Theorem. | [10] |
(Total = 38 hours. Add 5 hours for exams and leeway).
Last modified November 14, 2002