Ordered field where every nonnegative element is a square
This article is about ordered fields. For algebraic number fields whose ring of integers has a Euclidean algorithm, see Norm-Euclidean field. For the class of models in statistical mechanics, see Euclidean field theory.
In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K.
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield. In other words, the constructible numbers form the Euclidean closure of the rational numbers.
specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows...
as polymer field theory. In fact, by performing a Wick rotation from Minkowski space to Euclidean space, many results of statistical field theory can...
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements...
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}...
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest...
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...
to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic...
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a...
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...
vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . A vector field on...
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric...
mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number...
who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes...
(hence the name). In the case of Euclidean space, one usually defines the directional derivative of a vector field in terms of the difference between...
if the underlying metric is Manhattan distance. Common metrics are: Euclidean distance Taxicab geometry, also known as City block distance or Manhattan...
around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide...
computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers). Let F be a finite field. For any element x in F and...
Pythagorean closure. The Hilbert field is the minimal ordered Pythagorean field. Every Euclideanfield (an ordered field in which all non-negative elements...
bootstrap approach to conformal field theory in R d {\displaystyle \mathbb {R} ^{d}} . They are also referred to as Euclidean bootstrap axioms. Dirac–von...
turn is contained in the field of algebraic numbers. It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals...
In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity...
ring is isomorphic to Z under the dimension mapping. A formally real Euclideanfield E is not quadratically closed (as −1 is not a square in E) but the...
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