Construct an order on a field $\mathbb{Q}^2$ Let $\mathbb{Q}$ be the field of rational numbers. Let $\mathbb{Q}^2=\{(a,b):a,b\in\mathbb{Q}\}$ and define addition and multiplication as follows:
$$
(a,b)+(c,d)=(a+c,b+d)\\
(a,b)\cdot(c,d)=(ac+2bd, ad+bc)
$$
Then $(\mathbb{Q}^2, +, \cdot)$ is a field, and $(0,0)$ is its zero element while $(1,0)$ is its unit element. The inverse of $(a,b)$ is $(\frac{a}{a^2-2b^2},-\frac{b}{a^2-2b^2})$. I'm asked to construct an order on it such that it becomes an ordered field, namely invariant under addition and multiplication with a positive element.
What I know is $(1,0)>(0,0)$, so $(n,0)>(0,0)$ for any natural number $n$. I don't quite know how to proceed. Is there any rules to follow to construct an order on a field?
Thank you for any help!

The quick and easy answer is to notice that this field is isomorphic to $\mathbb Q[\sqrt{2}]$ by sending $(a,b)$ to $a+b\sqrt 2$, so $(a,b)<(c,d)$ if $a+b\sqrt 2<c+d\sqrt 2$.
