Algebra II

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Typical Scheduling: 
Every spring semester

Graduate level linear and abstract algebra including rings, fields, modules, some algebraic number theory and Galois theory. (2nd of two courses)

Course Text: 

Text at the level of Abstract Algebra, by Dummitt & Foote, 3rd edition, Wiley

Topic Outline: 
  • Intensive review of elementary ring theory: rings, homomorphisms, polynomial rings, ideals, quotients, integral domains, maximal and prime ideals
  • The prime spectrum of a ring, localization, the Zariski topology, the Nullstellensatz (brief remarks on varieties and algebraic geometry)
  • Integral domains, (unique) factorization, algebraic integers, ideal factorization and some ideal class theory (algebraic number theory). Important examples will include the integers, Gaussian integers, and the algebraic integers in imaginary quadratic fields
  • Geometry of numbers (time permitting)
  • Modules, generators and relations, structure theorem (Note: some of these ideas will have been introduced in the first semester in the section on abelian groups.)
  • Fields, algebraic field extensions, ruler and compass constructions, transcendental extensions, function fields algebraically closed fields (Note: finite fields are treated in the first semester.)
  • Galois theory: Galois extensions, Galois groups, cubic equations, symmetric functions and the discriminant, quartic equations, Kummer extensions, cyclotomic extensions, the irreducibility of a quintic