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Consider the field $$F = \{(a, b) | a, b \in \mathbb{R}\},$$ with two operations:

$$(a, b) + (c, d) = (a + c, b + d),$$

$$(a, b) · (c, d) = (ac − bd, ad + bc).$$

Prove that $F$ isn't an ordered field.

I've been trying to find ways this contradicts properties of ordered field, but couldn't find any. I'm also having difficulty defining what $(a,b) > (c,d)$ means.

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Do you recognize that this is just a very familiar field in disguise?

In any case, note that $(1, 0)$ is the unity of the field, and $(-1, 0)$ is its opposite. Note that $(0, 1)^{2} = (-1, 0)$.

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  • $\begingroup$ Do you mean that because (0,1)^2 is smaller than 0, then it violates the property that x^2>=0? How did you determine that (-1,0)<0? $\endgroup$ – blz Oct 26 '16 at 8:26
  • $\begingroup$ I am appealing to this. $\endgroup$ – Andreas Caranti Oct 26 '16 at 8:29

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