I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$).
I know just the $\tilde {K(x)}\subset K(x)$, I need help in the reverse implication.
I need help.
Thanks in advance.
I'll assume you're defining $\widetilde{K(u)}$ as follows. Given a field extension $K \subset L$ and $u \in L$, then $\widetilde{K(u)}$ is by definition the set of all elements of $L$ obtainable by some finite sequence of additions, multiplications, and divisions involving constants from $K$ and $u$.
In that case, to get the reverse implication for $\widetilde{K(x)} = K(x)$, pick some $p(x)/q(x) \in K(x)$ as usual and note that it can be obtained by some finite sequence of additions, multiplications, and divisions involving constants from $K$ and $u$. (I don't think any version of this question can be much less trivial.)