real closure of an archimedean field my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!

Since Archimedean fields and their real closures are just fields between $\Bbb Q $ and $\Bbb R$, and the rationals are already dense in the reals, the rationals must also be dense in the real closure of your field. Thus certainly the original field is dense in its real closure.