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    Integral Domains and Fields : Embedding Theorem

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    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.

    © BrainMass Inc. brainmass.com June 3, 2020, 5:11 pm ad1c9bdddf
    https://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.

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