Editing Gaussian rational

{{Short description|Complex number with rational components}}
In [[mathematics]], a '''Gaussian rational''' number is a [[complex number]] of the form ''p''&nbsp;+&nbsp;''qi'', where ''p'' and ''q'' are both [[rational number]]s.
The set of all Gaussian rationals forms the Gaussian rational [[field (mathematics)|field]], denoted '''Q'''(''i''), obtained by adjoining the [[imaginary number]] ''i'' to the field of rationals '''Q'''.

==Properties of the field==
The field of Gaussian rationals provides an example of an [[algebraic number field]] that is both a [[quadratic field]] and a [[cyclotomic field]] (since  ''i'' is a 4th [[root of unity]]).  Like all quadratic fields it is a [[Galois extension]] of '''Q''' with [[Galois group]] [[cyclic group|cyclic]] of order two, in this case generated by [[complex conjugation]], and is thus an [[abelian extension]] of '''Q''', with [[conductor (algebraic number theory)|conductor]] 4.<ref>[[Ian Stewart (mathematician)|Ian Stewart]], [[David O. Tall]], ''Algebraic Number Theory'', [[Chapman and Hall]], 1979, {{ISBN|0-412-13840-9}}. Chap.3.</ref>

As with cyclotomic fields more generally, the field of Gaussian rationals is neither [[ordered field|ordered]] nor [[complete space|complete]] (as a metric space). The [[Gaussian integer]]s '''Z'''[''i''] form the [[ring of integers]] of '''Q'''(''i''). The set of all Gaussian rationals is [[countable set|countably infinite]].

The field of Gaussian rationals is also a two-dimensional [[vector space]] over '''Q''' with natural [[Basis (linear algebra)|basis]] <math>\{1, i\}</math>.

==Ford spheres==
The concept of [[Ford circle]]s 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]], 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 <math>p/q</math> (i.e. {{tmath|p}} and {{tmath|q}} are relatively prime), the radius of this sphere should be <math>1/2|q|^2</math> where <math>|q|^2 = q \bar q</math> is the squared modulus, and {{tmath|\bar q}} is the [[complex conjugate]]. The resulting spheres are [[tangent]] for pairs of Gaussian rationals <math>P/Q</math> and <math>p/q</math> with <math>|Pq-pQ|=1</math>, and otherwise they do not intersect each other.<ref>{{citation|title=Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning|first=Clifford A.|last=Pickover|authorlink=Clifford A. Pickover|publisher=Oxford University Press|year=2001|isbn=9780195348002|contribution=Chapter 103. Beauty and Gaussian Rational Numbers|pages=243–246|url=https://books.google.com/books?id=52N0JJBspM0C&pg=PA243}}.</ref><ref>{{citation|year=2015|arxiv=1503.00813|title=Ford Circles and Spheres|first=Sam|last=Northshield|bibcode=2015arXiv150300813N}}.</ref>

==References==
{{reflist}}

[[Category:Cyclotomic fields]]

{{Numtheory-stub}}
{{Number systems}}

[[it:Intero di Gauss#Campo dei quozienti]]