As an example, consider the field of complex numbers. The field of complex numbers contains the field of real numbers as a subfield. However, the field of complex numbers does not have an order defined on it while the field of real numbers does.
I am wondering if there is an example of the reverse, where we start with an ordered field, and have a subfield of that ordered field which is not ordered.
(F,R) is an ordered field when F is a field and R is
a linear order of F adhering to a set of order axioms.
Let F' be a subfield of F. F' is not ordered.
(F',R') is an ordered subfield of (F,R) when
F' is a subfield of F and R' = R $\cap$ F'×F'.
So yes, subfields of an ordered field are not ordered
while ordered subfields inherits the field's order.
The question is really sematical, a matter of definitions.