Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?
Every natural number is the sum of four squares, and so you can define the positive rational numbers as those of the form $(a^2+b^2+c^2+d^2)/(e^2+f^2+g^2+h^2)$, where the denominator is not zero, and this is expressible in your language. And the order is defined by $x\lt y\iff yx$ is positive. 

