Question

Let R be an integral domain. Prove that if R is a field to begin with, then the field of quotients Q is isomorphic to R

Answer 1

firstly the way to go is to find Q inside of F and since Q is a field, its automatically a sub field of F. To understand why the problem is true, just think about it , what does any field containing R need to have? Well of course it needs all the elements of R, but since it needs to be a field, it also needs to contain all the inverses of elements in R But of course we can find these inverses in R itself since it's a field. Now, there is no guarantee this is all of F, F may have some extra stuff, but if we just take these out of F, we have Q! R's field of fractions.

View R as a sub ring of Q. Let F be a field and f:R→F a monomorphism. Then there is a unique isomorphism g:Q→F which extends f.

so the field of quotioents Q is isomorphic to R.