Difference between revisions of "Math 372: Abstract Algebra 2."

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=== Textbooks ===
=== Textbooks ===
Possible textbooks for this course include:
Possible textbooks for this course include (but are not limited to):
Joseph Rotman, ''Galois Theory (Second Edition)'', Springer, 1998.
Joseph Rotman, ''Galois Theory (Second Edition)'', Springer, 1998.

Revision as of 13:06, 24 May 2010

Catalog Information


Abstract Algebra 2.

(Credit Hours:Lecture Hours:Lab Hours)



F, W


Math 371.


Fields, Galois theory, solvability of polynomials by radicals.

Desired Learning Outcomes


Students to have mastered Math 313 (linear algebra) and Math 371 (group theory and ring theory) prior to enrolling in Math 372.

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.

  1. 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
  2. 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 probably including some of the material on rings.)

Ian Stewart, Galois Theory (Third Edition), Chapman Hall, 2004.

Additional topics

Courses for which this course is prerequisite