Handbook 1996 : Faculty of Science (Volume 4 page 213)
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618-321 Algebra

Note: To enter 618-321 a grade of H3 or better will normally be required in 618-222.

Credit points: 15.0

Coordinator: Dr S Gadde

Prerequisite: Mathematics 618-222(See Note below)

Contact: 39 lectures (three a week).

Timetable: First semester

Objectives:

On completion of this subject, students should:

Comprehend:

• the concepts of unique factorization domains;
• fields of fractions; modules;
• algorithmic nature of the structure theorem for modules over Principal Ideal Domains when specialized to Euclidean Domains;
• Galois correspondence; unsolvability in general of equations by radicals.

Have developed:

• the ability to find the structure of finitely generated abelian groups from their presentations;
• the ability to test polynomials of low degree for irreducibility; an understanding of the impossibility of trisecting an angle by ruler and compass;
• the ability to calculate Galois groups of equations in special cases.

Appreciate:

• the structure of special rings like Principal Ideal Rings;
• the possibility of relating problems in different areas by correspondences like Galois correspondence;
• that certain problems are not solvable and that it is possible to prove that they are not solvable in some interesting cases.

Content:

Modules over principal ideal domains: review of basic ring theory; ideals, quotients, the homomorphism theorems, prime and maximal ideals; integral domains and the field of quotients; Euclidean domains and principal ideal domains; definition and examples of modules; submodules, homomorphisms of modules, quotient modules; free modules and bases; structure of a finitely generated module over a principal ideal domain; applications to abelian groups. Field Theory: field extensions and their construction; the degree of a field extension; ruler and compass constructions; splitting fields; the Galois group of a field extension; the fundamental theorem of Galois theory.

Assessment:

Up to 26 pages of written assignments and up to three hours of end-of-semester written examination.

Mathematics subject : Next:618-322 | Prev:618-312 | Search | Help
Handbook 1996 : Faculty of Science (Volume 4 page 213)

Status:          Official 1996
Date created:    Oct  9 1995
Last modified:   Oct  9 1995
Authorised by:   Academic Registrar
Email enquiries: Course_Information@registrar.unimelb.edu.au

Copyright © University of Melbourne 1995,1996.