Math 372: Abstract Algebra 2.
Abstract Algebra 2.
(Credit Hours:Lecture Hours:Lab Hours)
Fields, Galois theory, solvability of polynomials by radicals.
Desired Learning Outcomes
This is a second course in abstract algebra focusing on field theory. The course is aimed at undergraduate mathematics majors, and it is strongly recommended for students intending to complete a graduate degree in mathematics. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.
- Ring Theory
- Basic Definitions
- Examples of rings (both commutative and noncommutative)
- Ideals and ring homomorphisms
- Quotient rings
- Prime and maximal ideals
- Polynomial rings over fields
- Factorization in polynomial rings
- Irreducible polynomials
- Field of fractions of a domain
- Field Theory
- Extensions of fields
- Field extensions via quotients in polynomial rings
- Automorphisms of fields
- Field of characteristic 0 and prime characteristic
- Galois extensions and Galois groups
- The Galois correspondence
- Independence of characters
- Fundamental Theorem of Galois Theory
- Fundamental Theorem of Algebra
- Roots of unity
- Solvability by radicals
- Insolvability of the quintic
Possible textbooks for this course include (but are not limited to):
Joseph Rotman, Galois Theory (Second Edition), Springer, 1998.
David Dummit and Richard Foote, Abstract Algebra (Third Edition), Wiley, 2003. (The chapters on fields and Galois theory, and some of the material on rings.)
Ian Stewart, Galois Theory (Third Edition), Chapman Hall, 2004.
The instructor may cover additional topics beyond the minimal requirements. Possible topics include (but are not limited to): introduction to algebraic numbers, applications of field extensions to cryptography, applications of field extensions to diophantine analysis, relations of field theory to algebraic geometry, construction of algebraically closed fields using Zorn's lemma, introduction to computer calculations in abstract algebra.