Is the mapping an identity map?

Question

F is a homomorphism from the field of real numbers into itself. I have proved that F is injective. I have also proved that F(q)=q for all 'q' belonging to the set of rational numbers. Is F the identity map? I am not able to come to a conclusion about images of irrational numbers under F. Can I get a few hints?

Answer 1

Hint: Once you've proven that $F(q)=q$ for rational $q$, it is sufficient to prove that $F$ preserves order, which is equivalent to proving that $F$ maps positive numbers to positive numbers.