Linear & Abstract Algebra & Number Theory
University of Queensland
Area of Study
Taught In English
Course Level Recommendations
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Host University Units2
Recommended U.S. Semester Credits4
Recommended U.S. Quarter Units6
Hours & Credits
OverviewCourse DescriptionThis course provides an introduction to the basics of linear and abstract algebra (including groups, rings & fields) & elementary number theory. Applications are included, enabling students to apply this knowledge in various fields.Course IntroductionMATH2301 provides an important foundation in number theory, abstract algebra, and linear algebra. Abstract algebra is further divided into groups, and rings & fields.An approximate break down by subject is as follows:Number Theory
- Divisibility, greatest common divisors, the Euclidean algorithm, the fundamental theorem of arithmetic, existence of infinitely many primes. Definition of integers mod n, inverses, Euler?s phi function, Chinese remainder theorem, Fermat?s little theorem, applications.
Rings & Fields
- Definition of groups and subgroups, basic properties. Examples: dihedral groups, matrix groups, symmetric groups etc. Cosets, Lagrange?s Theorem. Homomorphisms and Isomorphisms.
- Definition of rings, basic properties. Examples: matrix rings, polynomial rings etc. Zero divisors, units, integral domains, fields.
Learning ObjectivesAfter successfully completing this course you should be able to:
- Definition of an abstract vector space over an arbitrary field. Subspaces, bases, dimension. Inner products, orthogonal bases. Linear transformations, relations with matrices. Eigenvalues and eigenvectors, relationship with diagonalisation.
Class Contact3 Lecture hours, 1 Tutorial hourAssessment SummaryTutorial Quizes: 20%Problem Set 1: 10%Problem Set 2: 10%Final Exam: 60%
- solve simple number theory problems, including those that use the extended Euclidian algorithm, and the Chinese remainder theorem. You will be able to apply the Fundamental Theorem of Arithmetic, Euler's Theorem and Fermat's Little Theorem. You will also be able to find the order of an element and the inverse of an element mod n.
- solve abstract algebra problems, including being able to calculate kernels, cosets, the order of an element, the greatest common divisor of two elements of a polynomial ring, and Cayley tables. You should also be able to recall and apply definitions from abstract algebra in order to either prove or disprove that an object belongs to a particular class.
- define and work with abstract vector spaces over an arbitrary field; define and apply basic linear algebra concepts.
- apply the mathematical techniques of the course to simple problems from other disciplines.
- present clear and concise written solutions to a variety of mathematical problems.
- use internationally accepted standards of mathematical rigour and notation.
- discuss mathematical problems with their peers, and derive appropriate solutions.
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Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.