Examples of 'ordered field' in a sentence
Meaning of "ordered field"
An ordered field is a mathematical structure consisting of a set of elements along with two operations, addition and multiplication, that satisfy certain properties. The elements of an ordered field can be arranged in a total order, meaning that for any two elements a and b, either a < b, a = b, or a > b
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- A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.
How to use "ordered field" in a sentence
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ordered field
Every ordered field is a formally real field.
The real numbers are a complete ordered field.
An ordered field has some additional properties.
The reals are a complete ordered field.
Every other ordered field can be embedded in the surreals.
Real numbers as complete ordered field.
Every ordered field can be embedded in the surreal numbers.
The surreal numbers are the largest possible ordered field.
Every subfield of an ordered field is also an ordered field in the inherited order.
Its order and algebraic structure make it into an ordered field.
Let k be an ordered field.
The existence and uniqueness of the real closure of a countable ordered field.
Every ordered field contains an ordered subfield that is isomorphic to the rational numbers.
There are several ways of defining the real number system as an ordered field.
Hence the theory of the real ordered field with restricted analytic functions is model complete.
See also
The synthetic approach axiomatically defines the real number system as a complete ordered field.
Fact may be expressed by saying that the ordered field Q is isomorphic to the.
The synthetic approach gives a list of axioms for the real numbers as a complete ordered field.
This ordered field is not Archimedean.
This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.
Ordered field A field with a total order compatible with its operations.
Squares are necessarily non-negative in an ordered field.
Any Dedekind-complete ordered field is isomorphic to the real numbers.
However, the complex numbers are not an ordered field.
More generally, the substructures of an ordered field ( or just a field ) are precisely its subfields.
More generally, the definition applies to a vector space over an ordered field.
Otherwise, such field is a non-Archimedean ordered field and contains infinitesimals.
Essentially, our assumption is that the real numbers form an ordered field.
Main article, Ordered field.
Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field.
This is because any square in an ordered field is at least, but.
The same will hold for any non-Archimedean ordered field.
For example, it is not enough to construct an ordered field with infinitesimals.
Order - in short, build a complete ordered field.
Conversely, it is known that any weakly o-minimal ordered field must be real closed.
Now, remember, the reals are an ordered field.
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