Field of polynomials and field of fractions

Question

I have a very stupid question but really confused between the two symbols: can anybody make me understand the difference between the polynomial ring $R [X]$ and the other one $R(X)$ - I know the former one but confused with the later symbol. My second question is: in order to show that a field $K$ is the field of fraction of an integral domain $R$, is it enough to show that $K $ is the smallest field containing $R $ ?

Answer 1

Adressing the second question, let $U \subset F $ be any field containing $R$. Then the inverse of each $\dfrac{a}{1} \in R$ is in $U$, so, for any $a,b\in R$, $\dfrac{a}{1}\cdot\dfrac{1}{b} = \dfrac{a}{b} \in U$, thus $F = U$. We conclude that $F$ is the smallest field containing $R$.