MATH427 Advanced Algebra

Overview

Fundamental properties and foundational results of group theory and Galois theory.

Fundamental properties and foundational results of group theory and Galois theory.

About this paper

Restriction

MATH 401-412

Limited to

BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc

Contact

Mathematics 400-level programme co-ordinator: Dr Fabien Montiel 

Teaching staff

Dr Dominic Searles

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.

 

Textbooks

To be confirmed.

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.

Assessment details

3 written assignments (45% of total mark)

Final exam (55% of total mark)

Timetable