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