Relation between multiplicative and additive identity in an ordered field

Question

In class, we prove that 1>0 by using order axioms of a field. The question is can we conclude that multiplicative identity is always less than the additive identity for any arbitrary ordered field?

Answer 1

You have $-x \le 0 \le x$ or $x \le 0\le -x$, and $0\le x^2$ for every $x\in k$. Choose $x=1$. Now $x=x^2\neq0$, so $0<1$.