Gaussian rational

In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(i), obtained by adjoining the imaginary number i to the field of rationals.

Properties of the field

The field of Gaussian rationals provides an example of an algebraic number field, which 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.[1]

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.

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, 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 , the radius of this sphere should be where represents the complex conjugate of . The resulting spheres are tangent for pairs of Gaussian rationals and with , and otherwise they do not intersect each other.[2][3]

gollark: I mostly want an excuse to do stupid insane things.
gollark: What if CODE GUESSING 17259815?
gollark: What I can easily do is construct a backdoor which nobody else can use, but I don't think that qualifies.
gollark: And practical hidden flaws are more like "if you encrypt 2^16 bytes with the same key it is possible to determine some of the plaintext with slightly higher probability" or known plaintext attacks and such, rather than "hahaha any message whatsoever can be decrypted".
gollark: I have some rough ideas but they'd probably be obvious to anyone competent.

References
  1. Ian Stewart, David O. Tall, Algebraic Number Theory, Chapman and Hall, 1979, ISBN 0-412-13840-9. Chap.3.
  2. Pickover, Clifford A. (2001), "Chapter 103. Beauty and Gaussian Rational Numbers", Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning, Oxford University Press, pp. 243–246, ISBN 9780195348002.
  3. Northshield, Sam (2015), Ford Circles and Spheres, arXiv:1503.00813, Bibcode:2015arXiv150300813N.
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