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?