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

1. ## 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

2. 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 $\text{char}(D) = p$ 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 $\mathbb{Q}$.