Let $x$ be a fixed real number and let $\mathbb{Q}$ be the field of rational numbers. It is known that $\mathbb{R}$ is an extension field of $\mathbb{Q}$ and thus there exists some homomorphisms. Can we obtain the real number $x$ by a finite sequence of homomorphisms from an (appropriately chosen) rational number $q$?
Homomorphism takes the usual meaning that $$ f(a+b)=f(a)+f(b); \\ f(ab)=f(a)f(b);\\ f(1_{\mathbb{Q}})=1_{\mathbb{R}}. $$