Fundamental properties and foundational results of group theory and Galois theory.
About this paper
| Paper title | Advanced Algebra |
|---|---|
| Subject | Mathematics |
| EFTS | 0.0833 |
| Points | 10 points |
| Teaching period | Semester 1 (On campus) |
| Domestic Tuition Fees ( NZD ) | $620.00 |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Restriction
- MATH 401-412
- Limited to
- BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc
- Contact
Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz
- Teaching staff
- Paper Structure
Main topics (18 lectures, 50 min each):
Lectures 1-3
- External and internal direct products.
- P-groups.
- Fundamental Theorem of Finite Abelian Groups.
Lectures 4-5
- Simplicity of alternating group for n at least 5.
- (sub)normal and composition series.
- Solvability.
- Isomorphism theorems.
Lectures 6-11
- Group actions.
- Orbit-stabiliser.
- Class equation.
- Enumeration under group action.
- Polya’s theorem.
Lectures 12-14
- Sylow theorems and applications.
Lectures 13-18
- Field extensions.
- Minimal polynomial.
- Finite and simple extensions.
- Splitting fields.
- Structure of finite fields.
- Separable/normal/Galois extensions.
- Fundamental Theorem of Galois Theory.
- Insolvability of general quintic equations.
- Teaching Arrangements
18 lectures (50 minutes each).
Tutorials: Weekly drop-in sessions for help with assignments.
Assessment (45% of total mark): 3 written assignments.
Final exam (55% of total mark)
- Graduate Attributes Emphasised
Critical Thinking, Interdisciplinary Perspective, Lifelong Learning.
View more information about Otago's graduate attributes.- Learning Outcomes
On completion of the study of this paper, students are expected to:
- Understand fundamental approaches in abstract algebra.
- Know important properties of groups and fields.
- Understand how to rigorously prove theorems in abstract algebra.
- Understand the power of working with general algebraic concepts and applications of these including counting with to respect to symmetry.