Introduction To Rings And Fields the complete guide

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About Course

Introduction To Rings And Fields.

This course will cover the basics of abstract rings and fields, which are an important part of any abstract algebra course sequence. We will spend roughly 4-5 weeks on rings. We will begin with definitions and important examples. We will focus cover prime, maximal ideals, and important classes of rings like integral domains, UFDs and PIDs. We will also prove the Hilbert basis theorem about noetherian rings. The last 3-4 weeks will be devoted to field theory. We will give definitions, basic examples. Then we discuss the extension of fields, adjoining roots, and prove the primitive element theorem. Finally, we will classify finite fields.

INTENDED AUDIENCE: B.Sc and M.Sc students studying mathematics

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Course Content

Introduction To Rings And Fields

  • Introduction To Rings And Fields – Introduction
    00:00
  • Introduction, main definitions
    00:00
  • Examples of rings.
    00:00
  • More examples
    00:00
  • Polynomial rings 1
    00:00
  • Polynomial rings 2
    00:00
  • Problems 1
    00:00
  • Problems 2
    00:00
  • Problems 3
    00:00
  • Kernels, ideals
    00:00
  • Homomorphisms
    00:00
  • First isomorphism and correspondence theorems
    00:00
  • Examples of correspondence theorem
    00:00
  • Prime ideals
    00:00
  • Maximal ideals, integral domains
    00:00
  • Problems 4
    00:00
  • Problems 5
    00:00
  • Problems 6
    00:00
  • Field of fractions, Noetherian rings 1
    00:00
  • Noetherian rings 2
    00:00
  • Hilbert Basis Theorem
    00:00
  • Irreducible, prime elements
    00:00
  • Quotient rings
    00:00
  • Irreducible, prime elements, GCD
    00:00
  • Principal Ideal Domains
    00:00
  • Unique Factorization Domains 1
    00:00
  • Unique Factorization Domains 2
    00:00
  • Existence of maximal ideals
    00:00
  • Gauss Lemma
    00:00
  • Z[X] is a UFD
    00:00
  • Eisenstein criterion and Problems 7
    00:00
  • Problems 8
    00:00
  • Problems 9
    00:00
  • Field homomorphisms
    00:00
  • Algebraic elements form a field
    00:00
  • Degree of a field extension 1
    00:00
  • Degree of a field extension 2
    00:00
  • Field extensions 1
    00:00
  • Field extensions 2
    00:00
  • Splitting fields
    00:00
  • Finite fields 1
    00:00
  • Finite fields 2
    00:00
  • Finite fields 3
    00:00
  • Problems 10
    00:00
  • Problems 11
    00:00

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