a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

Constructing fields:

  • Constructing fields from rings
    • Field of fractions
    • Residue fields
  • Constructing fields within a bigger field
  • Field extensions
    • Algebraic extensions
    • Transcendence bases
  • Closure operations

Fields with additional structure

  • Ordered fields
  • Topological fields
  • Differential fields

Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication.

Invariants of fields:

  • Model theory of fields
  • The absolute Galois group
  • K-theory


  • Linear algebra and commutative algebra
  • Finite fields: cryptography and coding theory
  • Geometry: field of functions
  • Number theory: global fields
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