ordered field $\implies$ $0$ cannot be written as the sum of two squares

Question

I recently read in a forum post that "a necessary condition for a field to be ordered is that the number zero cannot be written as the sum of two squares". What does this mean? Why is this a necessary condition? What does the ability of two squares to sum to zero have to do with a field being ordered?

Answer 1

Remember, an ordered field isn't just a field with an ordering... it is a field with an ordering that respects the field's operations.

In particular: it must be the case that for any $a\in F$, $a^2\geq 0$.

Now, for any $a,b\neq0$, we must have $a^2>0$ and $b^2>0$; so, $a^2+b^2>0$.