I need the proof:
The set of Rational numbers has no proper sub-field.
Alternatively, if $\displaystyle F\subseteq\mathbb{Q}$ is a subfield then $\displaystyle 1\in F$, so that $\displaystyle 1+1,1+1+1,\ldots\in F$ so $\displaystyle \mathbb{N}\subseteq F$. But then $\displaystyle -1,-1-1,\ldots \in F$ so that $\displaystyle \mathbb{Z}\subseteq F$. But for each $\displaystyle 0\neq z\in\mathbb{Z}$ we have that $\displaystyle z^{-1} \in F$, so that $\displaystyle pq^{-1} \in F$ for any $\displaystyle p,q\in\mathbb{Z},q\neq 0$. Hence $\displaystyle \mathbb{Q}\subseteq F.$