# Field of quotients probs

• Apr 10th 2009, 03:16 AM
Zinners
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?
• Apr 10th 2009, 08:35 AM
NonCommAlg
Quote:

Originally Posted by Zinners
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.

Quote:

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 $g:D \to Q$ by $g(x)=\frac{x}{1}.$ show that g is a ring isomorphism.