On the Integral Representation of $a{x}^2+b{y}^2$ and the Artin Condition

Abstract

Given a number field $F$ with ${ø}_F$ its ring of integers. For certain $a,b$ and $\alpha$ in ${ø}_F$, we show that the Artin condition is the only obstruction to the local-global principle for integral solutions of equation $a{x}^2+b{y}^2=\alpha$. Some concrete examples are presented at last.

9 pages; git commit: 20140801a/2a08e8f