Is $F = \left\{ \frac{a}{2^i} : a,i \in \Bbb Z\right\}$ a field?

Question

Is $F = \left\{ \frac{a}{2^i} : a,i \in \Bbb Z\right\}\subset\mathbb Q$ a field?

I think that it is because:

For addition: $ \frac{a}{2^i} + \frac{-a}{2^i} = 0$ which is in $\mathbb{Q}$

For multiplication: $ \frac{a}{2^i} \times \frac{2^i}{a} = 1$ which the inverse in $\mathbb{Q}$

Hence $F$ is a field under $\mathbb{Q}$. Is my logic and answer correct? Thank you!

Answer 1

We have $\mathbb Z \subsetneq F \subsetneq \mathbb Q$. Now $\mathbb Q$ is the smallest field that contains $\mathbb Z$. Since $F\neq \mathbb Q$, $F$ cannot be a field.