Editing Gaussian rational (section)

==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>.