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 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 ra ...

Properties

* Every Euclidean field is an ordered Pythagorean field, but the converse is not true.Martin (1998) p. 89 * If ''E''/''F'' is a finite extension, and ''E'' is Euclidean, then so is ''F''. This "going-down theorem" is a consequence of the

Diller–Dress theorem 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 \lambd ...

.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 c ...

, 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 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

\mathbb

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

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...

\mathbb\cap\mathbb

is a Euclidean field. * The field of hyperreal numbers 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 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...

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. ...

.Martin (1998) p. 35 By the going-down result above, no

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

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 for ...

\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 close ...

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 of that is a Euclidean field and is an ordered extension of .

References

* * *

External links

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