Primitive generators of certain class fields (Q2347031)

Let \(n\) and \(N\) be positive integers, and \(p\) be an odd prime not dividing \(nN\). There exists a unique finite abelian extension \(L\) of the imaginary quadratic field \(K = \mathbb Q(\sqrt{-n})\), which is also Galois over \(\mathbb Q\), for which \(p =x^{2} + ny^{2}\) for some \(x\),\(y\) in \(\mathbb Z\) satisfying \(x= 1\pmod N\) and \(y = 0\pmod N\) if and only if \(p\) splits completely in \(L\). Using class field theory, \(p\) splits completely in \(L\) is equivalent to \((-n/p)=1\) and \(f(X)= 0\pmod p\) has an integer solution, where \(f(X)\in \mathbb{Z}[X]\) is the minimal polynomial of a real algebraic integer which generates \(L\) over \(K\). In this paper, the authors construct a primitive generator of such field \(L\) over \(K\) by the use of class field theory, complex multiplication and Shimura's reciprocity law. The authors give several concrete examples using the special values of primitive generators of the function fields of some modular curves.