# Module MA3411: Abstract algebra I

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2013-14
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. David Wilkins
Learning Outcomes
On successful completion of this module, students will be able to
• Justify with reasoned logical argument basic properties of polynomial rings and finite field extensions.
• Specify and justify with reasoned logical argument aspects of the relationships between polynomials with coefficients in some ground field, finite extensions of that ground field, and the groups of automorphisms of those field extensions.
• Determine the Galois groups of appropriately-chosen polynomials of low degree.
Module Content
The module will cover the following topics:
• Basic principles of group theory.
• Basic principles of ring theory.
• Basic properties of polynomial rings with coefficients in a field. Gauss's Lemma concerning products of primitive polynomials. Eisenstein's irreducibility criterion for polynomials.
• Basic properties of field extensions.
• The Tower Law.
• Basic properties of algebraic extensions. Proof that the degree of a simple algebraic extension is equal to the degree of the minimum polynomial of the adjoined element generating the extension.
• Solvability and Insolvability of Ruler and Compass Constructions.
• Splitting fields. Existence and isomorphism theorems for splitting fields. Normal extensions. Separability. Basic properties of finite fields. The Primitive Element Theorem. The Galois Group of a finite Field Extension. The Galois Correspondence.
• Procedures for determining the roots of quadratic, cubic and quartic polynomials from the coefficients of such polynomials.
• Galois groups of polynomials of low degree.
• The class equation of a finite group. Cauchy's theorem concerning the existence of elements of prime order in finite groups. Simple groups. Solvable groups.
• Galois's Theorem concerning the solvability of polynomial equations. A quintic polynomial that is not solvable by radicals.
Lecture notes, problems, worked solutions to selected past examination questions and further information relevant to the module are available from the module webpage at http://www.maths.tcd.ie/~dwilkins/Courses/MA3411/.
Module Prerequisite
Introduction to Group Theory (MA1214), or some equivalent module providing an introduction to Abstract Algebra. Also Fields, Rings and Modules (MA2215) is recommended, but not essential.
Assessment Detail
This module will be examined in a 2-hour examination in Trinity term. There is no continuous assessment.