Given a commutative Ring $R$ of ordered pairs $(x,y)$ of reals $x,y$ with addition and multiplication defined in the following way.
$$(x,y) + (u,v) = (x+u,y+v)$$ $$(x,y).(u,v) = (xu-yv,xv + yu)$$
I already showed that $R$ is an integral domain , now i need to show to prove $R$ is a field or not .
If $R$ is a field then every nonzero element $(x,y) \in R$ have a multiplicative inverse , which is $(x,y)(m,n) = (xm-yn,xn + ym) = (1,0)$
$(1,0)$ is the ring unity and $(m,n)$ is the multiplicative inverse and $xm - yn =1$, $xn + ym = 0$ how do we show that such $(m,n)$ exists or not ?