Ray class field extensions of real quadratic fields and solvability of congruences (Q1066949)

Let \(K={\mathbb{Q}}(\sqrt{d})\) be a real quadratic field with class number one and q a rational prime which splits in K, \(q\cong {\mathfrak pp}'.\) The ray class fields (quadratic extensions) of K with conductor \(f| (4){\mathfrak p}\infty_ 1\infty_ 2\) have been constructed and have been used for proving solvability criteria for congruences of biquadratic polynomials mod q. For example, if \(d=11\), than \(\chi^ 4-26\chi^ 2- 7\equiv 0\) mod q is solvable in \({\mathbb{Z}}\) if and only if \(q\equiv 1 mod 4\) and \(a\mp 5b\equiv 1,2,4\) mod 7 where \(q=a^ 2-11b^ 2\), \(a,b>0\), or \(q\equiv 3 mod 4\) and \(\pm a+5b\equiv 1,2,4\) mod 7, where \(q=-a^ 2+11b^ 2\), \(q\neq 7\), \(a,b>0\).