In mathematics, a Euclidean field is an

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...

for which every non-negative element is a square: that is, in implies that for some in . The

constructible number In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...

s 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 In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in . The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Eu ...

of the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...

Properties

* Every Euclidean field is an ordered

Pythagorean field In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...

, but the converse is not true.Martin (1998) p. 89 * If ''E''/''F'' is a finite

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...

, and ''E'' is Euclidean, then so is ''F''. This "going-down theorem" is a consequence of the Diller–Dress theorem.Lam (2005) p.270

Examples

* The real

constructible numbers In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is con ...

, those (signed) lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.Martin (1998) pp. 35–36 Every

real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...

is a Euclidean field. The following examples are also real closed fields. * The

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

\mathbb

with the usual operations and ordering form a Euclidean field. * The field of real algebraic numbers

\mathbb\cap\mathbb

is a Euclidean field. * The field of

hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...

s is a Euclidean field.

Counterexamples

* The

\mathbb Q

with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in

\mathbb Q

since the square root of 2 is

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

.Martin (1998) p. 35 By the going-down result above, no algebraic number field can be Euclidean. * The

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

\mathbb C

do not form a Euclidean field since they cannot be given the structure of an ordered field.

Euclidean closure

The Euclidean closure of an ordered field is an extension of in the

quadratic closure In mathematics, a quadratically closed field is a field in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples * The field of complex numbers is quadratically closed; more generally, any algebraically clos ...

of which is maximal with respect to being an ordered field with an order extending that of .Efrat (2006) p. 177 It is also the smallest subfield of the

algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...

of that is a Euclidean field and is an ordered

of .

References

* * *

External links

* {{PlanetMath, urlname=EuclideanField, title=Euclidean Field Field (mathematics)