>>10155194>>10155204Nope:

The field of rational numbers is the quotient field of the integral domain of integers.

One can prove the existence of a quotient field of an integral domain:

Let be an integral domain

Then there exists a quotient field of

(1) by proving that the for the inverse completion less zero of is an abelian group,

(2) by extending the operation on D to an addition of division products as: and proving that induceses the operation on its substructure D, so as to justify the use of + for both operations,

(3) by proving that is an abelian group

(4) and finally by proving that is a quotient field of

And one can also prove that this quotient field of is unique:

Let be quotient fields of integral domains respectively.

Let be a monomorphism

Then there is one and only one monomorphism extending , and:

Also, if is a ring isomorphism, then so is

(1) by proving that there is only one and only monomorphism extending

(2) and by proving that is an isomorphism if is.