# Integral Domains and Fields : Embedding Theorem

Problem:

Note: C is set containment

If R is an integral domain, show that the field of quotients Q in the Embedding Theorem is the smallest field containing R in the following sense:

If R C F, where F is a field, show that F has a sub-field K such that R C K and K is isomorphic to Q.

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#### Solution Preview

Proof:

R is an integral domain, it means R is a commutative ring with the multiplicative unit e. Suppose R C F, where F is a field. Now we construct a map f: Q->F. For any element x=a/b in Q, ...

#### Solution Summary

A proof involving fields is offered in the solution.

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