REPRESENTATIONS OF INTEGERS BY THE BINARY QUADRATIC FORM x(2) + xy plus ny(2)

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 is an element of N arbitrary) under extra conditions x equivalent to 1 mod m, y equivalent to 0 mod m on the variables. We also give some examples where their extended ring class numbers are less than or equal to 3.