Thread: Ingetral Domains and the Fund. Thm. of Homomorphisms

Let D be an integral domain, define phi: Z --> D by phi(n) = n*1, where Z is the set of integers and phi is a homomorphism.
If the char(D) = p, with p not equal to 0, show ker(phi)=pZ and if char(D)=0, show ker(phi)={0}
Using the fundamental theorem of homomorphisms, find a rind which is isomorphic to phi(Z) for both of the above cases.
Finally, let F be a field. If char(F)=p, and p not equal to 0, prove F contains a subfield isomorphic to Z_p.
If char(F)=0, then does F contains a subfield isomorphic to Q

Originally Posted by Coda202

Let D be an integral domain, define phi: Z --> D by phi(n) = n*1, where Z is the set of integers and phi is a homomorphism.
If the char(D) = p, with p not equal to 0, show ker(phi)=pZ and if char(D)=0, show ker(phi)={0}
Using the fundamental theorem of homomorphisms, find a rind which is isomorphic to phi(Z) for both of the above cases.
Finally, let F be a field. If char(F)=p, and p not equal to 0, prove F contains a subfield isomorphic to Z_p.
If char(F)=0, then does F contains a subfield isomorphic to Q

If then it means . Therefore, where is a multiple of . Thus, because this set consists of all multiples of . Therefore, . If then it means the only way is if and so which means . Now let , a field. If then by the above arguments is a subfield of which is isomorphic to . If then has a subdomain which is isomorphic to . But then it must contains its quotients, since is a field. Thus, a field of quotients of is embedded in , since is a field of quotients of it means contains a subfield isomorphic to .