Reference your last post Deveno. Your definitions are inconsistent- please correct. QXQ = ((q1,0) , (q2,0))? What does that have to do with (q,r)? Which member of your "field" is equal to sqrt2?
EDIT: Also, you have come up with the statement that (undefined symbol)^2 = 2
If you have a real number on the right you have to have a real number on the left. If your "field' contains that real number, what is it?