MATH6215 Algebra 2: Field extensions and Galois Theory

Offered By	Department of Maths
Academic Career	Graduate Coursework
Course Subject	Mathematics
Offered in	First Semester, 2010, Second Semester, 2010, and First Semester, 2011
Unit Value	6 units
Course Description	Just as there is a formula for solving a quadratic equation, there are similar formulae for solving the general cubic and quartic. Galois theory provides a solution to the corresponding problem for quintics --- there is no such formula in this case! Galois theory also enables us to prove (despite regular claims to the contrary) that there is no ruler and compass construction for trisecting an angle. Topics to be covered include: Galois Theory - fields Field extensions Normal extensions Separable extensions Revision of group theory, abelian and soluble groups Galois' Theorem Solubility of equations by radicals Finite fields Cyclotomic fields Note: Graduate students attend joint classes with undergraduates but will be assessed separately.
Learning Outcomes	On satisfying the requirements of this course, students will have the knowledge and skills to: 1. Explain the fundamental concepts of field extensions and Galois theory and their role in modern mathematics and applied contexts 2. Demonstrate accurate and efficient use of field extensions and Galois theory 3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from field extensions and Galois theory 4. Apply problem-solving using field extensions and Galois theory applied to diverse situations in physics, engineering and other mathematical contexts.
Indicative Assessment	Assessment will be based on: Assignment 1 (20%; LO 1-4) Assignment 2 (20%; LO 1-4) Assignment 3 (20%; LO 1-4) Final exam (40%; LO 1-4)
Workload	36 lectures, tutorials by arrangement
Course Classification(s)	AdvancedAdvanced courses are designed for students having reached 'first degree' level of assumed knowledge, which provide a deep understanding of contemporary issues; or 'second degree' and higher levels of knowledge; or for transition to research training programs. and SpecialistSpecialist courses are designed for students having reached 'first degree' level of assumed knowledge, which provide for the acquisition of specialist skills; or 'second degree' and higher level of knowledge; or for transition to research training programs; or knowledge associated with professional accreditation.
Areas of Interest	Mathematics
Eligibility	Bachelor degree; with second year Mathematics
Requisite Statement	Second year Mathematics is required.
Programs	Master of Mathematical Sciences

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