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MATH 3158

Rings and Fields (Honours)

0.5 credit · Undergraduate

Rings, integral domains, Euclidean and principal ideal domains, fields, polynomial rings over a field, algebraic extensions of fields, the fundamental theorem of Galois theory, finite fields, applications.

2021

Topics

  • Introduction, algebraic elements
  • algebraic elements and extension fields, adjoining roots
  • degree of field extensions, algebraic extensions
  • Feb. 4 finite fields; number of irreducible polynomials.
  • transcendental extension, transcendence basis
  • primitive elements and primitive element theorem
  • Mar. 4 extension of a homomorphism, finite normal extension
  • Galois group of finite extension, finite normal extension
  • Galois correspondence theorem
  • cyclotomic polynomials and regular n-gon
  • Apr. 1 Solvable groups and solvable extensions
  • example of unsolvable quintic equation; # of real roots
  • Extensions of finite fields.

Readings

  • “Algebra.”, by Michael Artin, Prentice Hall, 2nd edition, 2010.
  • “Galois Theory”, by David A. Cox, Wiley, 2nd edition, 2012.
  • “Algebra with Galois Theory”, by Emil Artin, New York University, Ameri- can Mathematical Society, 2007.
  • “Field Theory and Galois Theory”, by Steven H. Weintraub, Universitext, Springer Science & Business Media, Inc., 2006.
  • “Fields and Galois Theory”, by John M. Howie, Springer Undergraduate Mathematics Series 2006.

Prerequisites & course information

Precludes additional credit for MATH 3108 . Prerequisite(s): MATH 2100 with a grade of C- or higher, or ( MATH 2108 or MATH 3101 with a grade of B or higher and MATH 1800 with a grade of B or higher and permission of the instructor), or permission of the School. Lectures three hours a week, tutorial one hour a week.

Sources