REPRESENTATIONS OF INTEGERS BY THE BINARY QUADRATIC FORM

Abstract

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.