Editing Complex number (section)

===Algebraic number theory===
[[File:Pentagon construct.gif|right|thumb|Construction of a regular pentagon [[compass and straightedge constructions|using straightedge and compass]].]]
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in <math>\mathbb{C}</math>. ''[[Argumentum a fortiori|A fortiori]]'', the same is true if the equation has rational coefficients. The roots of such equations are called [[algebraic number]]s – they are a principal object of study in [[algebraic number theory]]. Compared to <math>\overline{\mathbb{Q}}</math>, the algebraic closure of <math>\mathbb{Q}</math>, which also contains all algebraic numbers, <math>\mathbb{C}</math> has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of [[field theory (mathematics)|field theory]] to the [[number field]] containing [[root of unity|roots of unity]], it can be shown that it is not possible to construct a regular [[nonagon]] [[compass and straightedge constructions|using only compass and straightedge]] – a purely geometric problem.

Another example is the [[Gaussian integer]]s; that is, numbers of the form {{math|''x'' + ''iy''}}, where {{mvar|x}} and {{mvar|y}} are integers, which can be used to classify [[Fermat's theorem on sums of two squares|sums of squares]].