The one to concentrate on will be multiplicative inverses.

What is the inverse of

with

and

rationals?

What if both were integers, does that imply that the inverse has integer coefficients? (Alternatively, think about what the problem is saying...take

. Then if your ring was a field every element in this subset has an inverse in the ring. This subset is just the integers. So, every integer has an inverse of the form

. That is, every rational number of the form

can be written as

. As we are, allegedly, in a field we have that

is also of this form, with

. Thus, every rational number is of the form

. This is silly!)