MATH2712 Algebra

2016

4.5

1 x 120-minute lecture-2 weekly
1 x 60-minute lecture-1 weekly
1 x 50-minute tutorial weekly

1 1 of MATH1121, MATH1141
2 1 of MATH1122, MATH1142
Must Satisfy: (1 and 2)

1 of MATH2035, MATH3731, MATH8731 has been successfully completed

Equivalence relations, groups, subgroups, isomorphism of groups, transformation groups, realization of a group as a transformation group, cyclic groups, order of an element, invariant subgroups, factor groups, homomorphism of groups, the fundamental theorem of homomorphism for groups, abelian groups, classification of finitely generated abelian groups; linear representations of groups, irreducible representations, characters of representations; rings, commutative rings, integral domains, division rings, quarternions, fields, matrix rings, polynomial rings, subrings, center of a ring, ideals, quotient rings, ideals and quotient rings for the ring of integers, homomorphism of rings, fundamental theorem of homomorphisms of rings, simple ideals, prime ideals, principal ideals, coordinate ring of an algebraic curve, Hilbert basis theorem, the Hilbert Nullstellensatz; fields, vector spaces over a field, field of fractions of a commutative integral domain, extensions of fields, simple extensions, subfields, prime fields, the structure of finite fields, the field of rational functions on an algebraic curve; Galois theory; basics of homological algebra and category theory: categories, functors, commutative diagrams, modules, projective and injective modules.

This topic introduces basic notions of modern algebra.

At the completion of the topic, students are expected to be able to:

1. Have a proper knowledge of modern abstract algebra
2. Apply methods of abstract algebra to solution of practical problems
3. Understand the science behind reliability of moderncommunication systems and the security of bank transactions
4. Get a considerable level of abstract thinking
5. Understand proofs and be able to prove mathematical statements
6. Better understand the role and significance of mathematics in the modern world