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Math Help - Field of quotients probs

  1. #1
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    Field of quotients probs

    Hi,

    Can anyone get me going on these two problems below. Not really sure how to start off. Thanks

    The ring Z6 cannot be imbedded in a field. Why?

    The field of quotients of any field D is isomorphic to D. Why?
    Last edited by Zinners; April 10th 2009 at 06:36 AM.
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  2. #2
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    Quote Originally Posted by Zinners View Post
    Hi,

    Can anyone get me going on these two problems below. Not really sure how to start off. Thanks

    The ring Z6 cannot be imbedded in a field. Why?
    because it's not a domain: suppose there's a ring monomorphism f: \mathbb{Z}/6 \to F, where F is a field. we have f(2)f(3) = f(6) = f(0) = 0. but F is a domain and thus either f(2) = 0 or f(3) = 0.

    since f is one-to-one, we get either 2 = 0 or 3 = 0 in \mathbb{Z}/6, which is nonsense.

    The field of quotients of any field D is isomorphic to D. Why?
    let Q be the quotient field of a field D. define the map \to Q" alt="g \to Q" /> by g(x)=\frac{x}{1}. show that g is a ring isomorphism.
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