MATH-215 / 5 credits

Teacher: Monin Leonid

Language: English


Summary

This is an introductory course in ring and field theory.

Content

Fundamental concepts, constructions, and theorems:

  • Rings, subrings, homomorphisms of rings
  • Examples of rings
  • Integral rings, fields of fractions
  • Ideals, quotient rings and their universal properties, the characteristic of a ring, operations on ideals, correspondence theorems, product of rings, the Chinese remainder theorem
  • Prime and maximal ideals

Arithmetics in rings

  • Euclidean rings
  • Principal rings
  • Associated, prime, and irreducible elements
  • Factorial rings
  • Noetherian rings
  • Characterization of factoriality
  • Gauss's Lemma and theorem
  • Irreducibility criteria

Field Theory:

  • Algebras over a field
  • Field extensions, algebraic and transcendental elements, the degree of a field extension, algebraic extensions, construction of simple field extensions, splitting fields
  • finite fields
  • separable extensions, primitive element theorem
  • Galois theory
  • purely inseparable extensions, separable-inseparable decomposition
  • algebraically closed fields, separable closure, inseparable closure

Learning Prerequisites

Required courses

  • Algebraic structures
  • Linear algebra I and II
  • Group theory

 

Assessment methods

Written exam

Resources

Moodle Link

In the programs

  • Semester: Spring
  • Exam form: Written (summer session)
  • Subject examined: Algebra III - rings and fields
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: mandatory

Reference week

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