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Rings and Fields: Course Outline (MAT 451)

The prototype example is the ring of integers with the two operations of addition and multiplication. The rational, real and complex numbers are commutative rings of a type called fields. An algebra over a ring is itself a ring.

Course Objective

At the end of this course the students will be able to understand the ring, polynomial rings, Cartesian product of sets, direct product and direct sum of modules, HomR(M,N), algebra over a commutative ring, Euclidean Domains and analogous theorems for Fields with applications. 

Course Contents

  • Definitions and basic concepts,
  • Homomorphisms, Homomorphism theorems,
  • Polynomial rings,
  • Unique factorization domain,
  • Factorization theory,
  • Euclidean domains,
  • Arithmetic in Euclidean domains,
  • Extension fields,
  • Algebraic and transcendental elements,
  • Simple extension,
  • Introduction to Galois theory.

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