Is the set of dyadic rationals a field? I recently learned that the dyadic rationals is the set of rational numbers of the form $$\frac{p}{2^q}$$ where $p$ is an integer and $q$ is greater than or equal to zero. I think the set of dyadic rationals is not a field. Here's why: One of the requirements for a set to be a field is this: Every element $a$ in the set has exactly one reciprocal such that $a$ multiplied by the reciprocal equals $1$. I think that this dyadic rational does not have a reciprocal: $$\frac34$$ The reciprocal is $4/3$, but that is not a dyadic rational because there is no integer $q$ greater than or equal to $0$ such that $2^q=3$. Therefore the dyadic rationals fails one of the requirements for being a field. Therefore the dyadic rationals are not a field. Ha! How about that logic. Am I thinking correctly? Am I correct?

Correct.

The _ring_ of dyadic rationals is obtained from the integers $\Bbb Z$ inverting $2$. But that is not enough to invert all non-zero integers.

Besides, there's no smaller field of the rationals $\Bbb Q$ containing $\Bbb Z$.