I am trying to prove that the set $$\mathbb{Q}(\sqrt{2}) = \{a + b*\sqrt2\ \mid a,b \in \mathbb{Q} \}$$ is an ordered field. Proving that this is a set is easy but "order" part was a challenge for me. In the book I am reading (Introduction to Mathematical Structures by Steven Galovich), the ordered field axioms are as follows:
$(i)$ If $x$ and $y$ are in $F^+$, then $x + y$ and $x*y$ are also in $F^+$
$(ii)$ For each $x \in F$, exactly one of the following three conditions holds: $$(a)\;x \in F^+,$$ $$(b) \; x = 0,$$ $$(c) \; (-x) \in F^+ $$
I tried various criteria for $F^+$ however none of them worked. Can anyone show a suitable criteria or give an indirect proof for this set being an ordered field. (Except for the one that use "$R$ is ordered and this is a subset...", because that's beyond my current knowledge).
Hint: $\mathbb{Q}(\sqrt{2})\subseteq \mathbb{R}$, which is an ordered field.