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 of, types and/or fields:
“Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
—Talcott Parsons (1902–1979)
“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)
“We shall fight on the beaches, we shall fight on the landing grounds, we shall fight in the fields and in the streets, we shall fight in the hills; we shall never surrender.”
—Winston Churchill (1874–1965)