Glossary of Field Theory - Types of Fields

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.

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