Complete field
Encyclopedia
In mathematics
, a complete field is a field
equipped with a metric
and complete with respect to that metric. Basic examples include the real number
s, the complex number
s, and complete valued fields (such as the p-adic numbers).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a complete field is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
equipped with a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
and complete with respect to that metric. Basic examples include the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, and complete valued fields (such as the p-adic numbers).
See also
- Completion (ring theory)Completion (ring theory)In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
- Hensel's lemmaHensel's lemmaIn mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...
- Henselian ringHenselian ringIn mathematics, a Henselian ring is a local ring in which Hensel's lemma holds. They were defined by , who named them after Kurt Hensel.Some standard references for Hensel rings are , , and .-Definitions:...
- Ostrowski's theoremOstrowski's theoremOstrowski's theorem, due to Alexander Ostrowski , states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.- Definitions :...