Editing Complex multiplication (section)

==Sample consequence==

It is no accident that [[Ramanujan's constant]], the [[transcendental number]]<ref>{{MathWorld|title=Transcendental Number|urlname=TranscendentalNumber}} gives <math>e^{\pi\sqrt{d}}, d \in Z^*</math>, based on
Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974.</ref>

: <math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007\dots\,</math>
or equivalently,

: <math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007\dots\,</math>

is an [[almost integer]], in that it is [[Mathematical coincidence#Containing pi or e and number 163|very close]] to an [[integer]].<ref>[http://mathworld.wolfram.com/RamanujanConstant.html Ramanujan Constant – from Wolfram MathWorld<!-- Bot-generated title -->]</ref> This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of [[modular forms]], and the fact that
: <math>\mathbf{Z}\left[ \frac{1+\sqrt{-163}}{2}\right]</math>
is a [[unique factorization domain]].

Here <math>(1+\sqrt{-163})/2</math> satisfies {{nowrap|1=''α''<sup>2</sup> = ''α'' &minus; 41}}. In general, ''S''[''α''] denotes the set of all [[polynomial]] expressions in α with coefficients in ''S'', which is the smallest ring containing ''α'' and ''S''. Because α satisfies this quadratic equation, the required polynomials can be limited to degree one.

Alternatively,

: <math>e^{\pi \sqrt{163}} = 12^3(231^2-1)^3+743.99999999999925007\dots\,</math>

an internal structure due to certain [[Eisenstein series]], and with similar simple expressions for the other [[Heegner number]]s.