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Math Help - Ingetral Domains and the Fund. Thm. of Homomorphisms

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    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
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    Quote Originally Posted by Coda202 View Post
    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 \text{char}(D) = p then it means 1+...+1 = p\cdot 1 = 0. Therefore, k\cdot 1 = 0 where k is a multiple of p. Thus, \ker (\phi) = p\mathbb{Z} because this set consists of all multiples of p. Therefore, \phi (\mathbb{Z}) \simeq \mathbb{Z}/p\mathbb{Z} = \mathbb{Z}_p. If \text{char}(D) = 0 then it means the only way k\cdot 1 = 0 is if k=0 and so \ker (\phi) = \{ 0 \} which means \phi(Z) \simeq \mathbb{Z}/\{0\} \simeq \mathbb{Z}. Now let D=F, a field. If \text{char}(F) = p then by the above arguments \phi(\mathbb{Z}) is a subfield of F which is isomorphic to \mathbb{Z}_p. If \text{char}(F) = \{0\} then F has a subdomain \phi(\mathbb{Z}) which is isomorphic to \mathbb{Z}. But then it must contains its quotients, since F is a field. Thus, a field of quotients of \mathbb{Z} is embedded in F, since \mathbb{Q} is a field of quotients of \mathbb{Z} it means F contains a subfield isomorphic to \mathbb{Q}.
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