# About to lose it: rings, polynomials, and ideals.

For your specific question, the set $\{(m\cdot1_F)(n\cdot1_F)^{-1}\,:\,m,n\in\mathbb Z\,\text{with}\,n\ne0\}$ is easily shown to be a subfield of F. ( $n\cdot1_F)^{-1}$ exists since F has characteristic 0). Next the map:
$f\,:\,(m\cdot1_F)(n\cdot1_F)^{-1}\mapsto{m\over n}$ is an isomorphism onto Q -- you have to show f is well defined and then is an isomorphism, grunge work but easy.