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Let R be an integral domain and F a field such that R is a subring of F, and every element of F is
equal to the quotient of two elements of R. Show that F is isomorphic to Q(R) (the field of
quotients of R).
Solution
If x belongs to F, then x is of the form p/q where p and q are elements of R.
Thus let p/q be of the form simplest or p and q have no common factor between them.
Then each x can be expressed uniquely as p/q form.
Thus there exists a bijective mapping from F to Q(R)
Hence F is isomorphic to Q(R)

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  • 1. Let R be an integral domain and F a field such that R is a subring of F, and every element of F is equal to the quotient of two elements of R. Show that F is isomorphic to Q(R) (the field of quotients of R). Solution If x belongs to F, then x is of the form p/q where p and q are elements of R. Thus let p/q be of the form simplest or p and q have no common factor between them. Then each x can be expressed uniquely as p/q form. Thus there exists a bijective mapping from F to Q(R) Hence F is isomorphic to Q(R)