Gaussian rational

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 of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.Ian Stewart, David O. Tall, Algebraic Number Theory, Chapman and Hall, 1979, {{ISBN|0-412-13840-9}}. Chap.3.

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

The field of Gaussian rationals is also a two-dimensional vector space over Q with natural basis $\backslash \{1,\; i\backslash \}$.

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, 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 $p/q$ (i.e. {{tmath|p}} and {{tmath|q}} are relatively prime), the radius of this sphere should be $1/2|q|^2$ where $|q|^2\; =\; q\; \backslash bar\; q$ is the squared modulus, and {{tmath|\bar q}} is the complex conjugate. The resulting spheres are tangent for pairs of Gaussian rationals $P/Q$ and $p/q$ with $|Pq-pQ|=1$, and otherwise they do not intersect each other.{{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}}.{{citation|year=2015|arxiv=1503.00813|title=Ford Circles and Spheres|first=Sam|last=Northshield|bibcode=2015arXiv150300813N}}.

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Category:Cyclotomic fields

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it:Intero di Gauss#Campo dei quozienti