Types of Fields
- Finite field
- A field with finitely many elements.
- Ordered field
- A field with a total order compatible with its operations.
- Rational numbers
- Real numbers
- Complex numbers
- Number field
- Finite extension of the field of rational numbers.
- Algebraic numbers
- The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory.
- Quadratic field
- A degree-two extension of the rational numbers.
- Cyclotomic field
- An extension of the rational numbers generated by a root of unity.
- Totally real field
- A number field generated by a root of a polynomial, having all its roots real numbers.
- Formally real field
- Real closed field
- Global field
- A number field or a function field of one variable over a finite field.
- Local field
- A completion of some global field (w.r.t. a prime of the integer ring).
- Complete field
- A field complete w.r.t. to some valuation.
- Pseudo algebraically closed field
- A field in which every variety has a rational point.
- Henselian field
- A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields.
Read more about this topic: Glossary Of Field Theory
Famous quotes containing the words types and/or fields:
“The bourgeoisie loves so-called “positive” types and novels with happy endings since they lull one into thinking that it is fine to simultaneously acquire capital and maintain one’s innocence, to be a beast and still be happy.”
—Anton Pavlovich Chekhov (1860–1904)
“It’s a melody full of the laughter of children out after the rain.”
—Dorothy Fields (1904–1974)