# Integral Domains and Fields : Embedding Theorem

Not what you're looking for? Search our solutions OR ask your own Custom question.

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.

Â© BrainMass Inc. brainmass.com March 4, 2021, 5:54 pm ad1c9bdddfhttps://brainmass.com/math/integrals/integral-domains-fields-embedding-theorem-17278

#### 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.

$2.49