Chapter 7
Rings and fields


  1. The structure ring

  2. Constructions with rings

  3. Domains and fields

  4. Fields

  5. Ideals

  6. Quotient rings

  7. Finite fields

  8. Exercises

  9. Summary of Chapter 7

We continue the study of structures. Having done monoids and groups in the previous chapter, we now concentrate on two important structures in which they play a significant role: rings and fields.

Section 7.1
The structure ring
  1. Rings and subrings
  2. Some computational laws
  3. Invertible elements
  4. Morphisms
Section 7.2
Constructions with rings
  1. Cartesian products
  2. Generation
  3. Polynomial rings
Section 7.3
Domains
  1. The notion
  2. More on domains
  3. Fields
  4. Subfields
  5. Field of fractions
Section 7.4
Fields
  1. Characteristics
  2. Vector spaces
  3. General properties
  4. Morphisms
  5. Algebraic numbers
Section 7.5
Ideals
  1. The notion
  2. Generation
  3. Constructions
  4. Prime and maximal ideals
Section 7.6
Quotient rings
  1. Residue class rings
  2. First isomorphism theorem
  3. Prime and maximal ideals
Section 7.7
Finite fields
  1. Characterization
  2. Primitive elements
  3. Existence
  4. Hadamard matrices