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

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