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?

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- Apr 10th 2009, 03:16 AMZinnersField 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? - Apr 10th 2009, 08:35 AMNonCommAlg
because it's not a domain: suppose there's a ring monomorphism $\displaystyle f: \mathbb{Z}/6 \to F,$ 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 $\displaystyle \mathbb{Z}/6,$ which is nonsense.

Quote:

The field of quotients of any field D is isomorphic to D. Why?