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Construct an order on a field $\mathbb{Q}^2$

Discussion in 'Mathematics' started by Edward Wang, Oct 8, 2018.

  1. Edward Wang

    Edward Wang Guest

    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!

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