MATH3731 Algebra

This topic includes: Basic concepts of mathematical logic and set theory. Axiomatic definition of integers. Divisibility, primes, quotient-remainder theorem, Euclidean algorithm, co-primes, Euclid's lemma, prime factorisation. Congruence classes and modular arithmetic. Groups, subgroups, cyclic groups, order of a group element, generators, Euler's function. Permutations, even and odd, definition of determinant. Homomorphisms and isomorphisms of groups, Cayley's theorem. Cosets, Lagrange's theorem. Normal subgroups, quotient groups, first isomorphism theorem. Cartesian product of groups. RSA public key cryptography. Rings, subrings, units, zero-divisors, integral domains and fields. Polynomial rings over fields, remainder and factor theorems. Irreducible and coprime polynomials. Euclid's lemma and prime factorisation for polynomials. Roots of polynomials, criteria of irreducibility. Homomorphism and isomorphisms of rings. Ideals, principal ideals, quotient rings. Prime and maximal ideals. Ideals of polynomial rings, finite fields and their construction.

This topic aims to provide:

• A better understanding of specificity of mathematics as a deductive science based on pure logic and exposure to a significant amount of abstract mathematical thinking
• A deep understanding of integers, divisibility, primes and their properties
• Knowledge of the basic theory of groups and their applications
• Knowledge of the basic theory of rings and their applications
• Better understanding of mathematical concepts in general and of mathematical way of thinking
• Knowledge which will assist in better understanding and learning of other mathematics, physics and engineering topics

1. Demonstrate deep knowledge of divisibility and other associated properties of integers
2. Demonstrate better understanding of logical and deductive structure of mathematics and understand and communicate proofs of mathematical statements
3. Demonstrate abstract mathematical thinking
4. Interpret symmetries of physical and engineering systems using group theory methods
5. Discuss the kind of mathematics on which reliability of modern communication systems rests
6. Demonstrate better understanding the role and significance of mathematics in the modern world