Introduction To Rings And Fields the complete guide

  • Course level: Beginner
  • Categories C-Science
  • Last Update 23/06/2021


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

Topics for this course

45 Lessons

Introduction To Rings And Fields

Introduction To Rings And Fields – Introduction00:00:00
Introduction, main definitions00:00:00
Examples of rings.00:00:00
More examples00:00:00
Polynomial rings 100:00:00
Polynomial rings 200:00:00
Problems 100:00:00
Problems 200:00:00
Problems 300:00:00
Kernels, ideals00:00:00
First isomorphism and correspondence theorems00:00:00
Examples of correspondence theorem00:00:00
Prime ideals00:00:00
Maximal ideals, integral domains00:00:00
Problems 400:00:00
Problems 500:00:00
Problems 600:00:00
Field of fractions, Noetherian rings 100:00:00
Noetherian rings 200:00:00
Hilbert Basis Theorem00:00:00
Irreducible, prime elements00:00:00
Quotient rings00:00:00
Irreducible, prime elements, GCD00:00:00
Principal Ideal Domains00:00:00
Unique Factorization Domains 100:00:00
Unique Factorization Domains 200:00:00
Existence of maximal ideals00:00:00
Gauss Lemma00:00:00
Z[X] is a UFD00:00:00
Eisenstein criterion and Problems 700:00:00
Problems 800:00:00
Problems 900:00:00
Field homomorphisms00:00:00
Algebraic elements form a field00:00:00
Degree of a field extension 100:00:00
Degree of a field extension 200:00:00
Field extensions 100:00:00
Field extensions 200:00:00
Splitting fields00:00:00
Finite fields 100:00:00
Finite fields 200:00:00
Finite fields 300:00:00
Problems 1000:00:00
Problems 1100:00:00
Introduction To Rings And Fields

Enrolment validity: Lifetime


  • A little bit of abstract group theory and a little bit of linear algebra.

Target Audience

  • Week 1: Definition of rings, examples, polynomial rings, a homomorphism.
  • Week 2: Ideals, prime and maximal ideals, quotient rings.
  • Week 3: Noetherian rings, Hilbert basis theorem.
  • Week 4: Integral domains, quotient fields.
  • Week 5: Unique factorization domains, principal ideal domains.
  • Week 6: Definition of fields, examples, degree of field extensions.
  • Week 7: Adjoining roots, primitive element theorem.
  • Week 8: Finite fields.