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.