mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Gaussian rational number is a

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 ...

of the form ''p'' + ''qi'', where ''p'' and ''q'' are both

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 ...

s. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained by adjoining the

imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...

''i'' to the field of rationals.

Properties of the field

The field of Gaussian rationals provides an example of an

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 f ...

, which is both a quadratic field and a cyclotomic field (since ''i'' is a 4th

root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...

). Like all quadratic fields it is a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...

of Q with

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

of order two, in this case generated by

complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

, and is thus an abelian extension of Q, with conductor 4. As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

(as a metric space). The Gaussian integers Z 'i''form the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

of Q(''i''). The set of all Gaussian rationals is

countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

Ford spheres

The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as

p/q

, the radius of this sphere should be

1/q\bar q

where

\bar q

represents the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

q

. The resulting spheres are

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

for pairs of Gaussian rationals

P/Q

and

p/q

with

, Pq-pQ, =1

, and otherwise they do not intersect each other..

References

Cyclotomic fields {{Numtheory-stub it:Intero di Gauss#Campo dei quozienti