TITLE: In an ordered field, must 1 be positive?
QUESTION [7 upvotes]: In an ordered field, must the multiplicative identity be positive? Or must it be defined as such?

REPLY [13 votes]: Remember that for a total order,

If $a \leq b$, then $a+c \leq b+c$.
If $0 \leq a$ and $0 \leq b$, then $0 \leq ab$.

If $1 \leq 0$, then $1+(-1) \leq 0 + (-1)$ i.e. $0 \leq -1$. By ($2$), we need $0 \leq (-1)(-1) = 1$.
Hence, we get that $1 \leq 0 \leq 1$. For a non-trivial field, $0 \neq 1$. Hence, we get a contradiction that $$1 < 0 < 1.$$
Hence, $0 < 1$.