# Thread: Subfield Isomorphic To Q

1. ## Subfield Isomorphic To Q

I would like to know how to show that if the characteristic of a field is 0 then it has a subfield isomorphic to $\mathbb Q$.

I defined an homomorphism $f : \mathbb F \rightarrow \mathbb Q$
$f(n) = n \cdot 1$
I found, $Ker(f) = \{0\}$ The map is thus injective.
I thought I might find a surjective map to $\mathbb F/\{0\}$ and use the first isomorphism theorem to conclude that $\mathbb F/\{0\}$ is a subfield isomorphic to $\mathbb Q$.

2. Originally Posted by vincisonfire
I would like to know how to show that if the characteristic of a field is 0 then it has a subfield isomorphic to $\mathbb Q$.

I defined an homomorphism $f : \mathbb F \rightarrow \mathbb Q$
$f(n) = n \cdot 1$
I found, $Ker(f) = \{0\}$ The map is thus injective.
I thought I might find a surjective map to $\mathbb F/\{0\}$ and use the first isomorphism theorem to conclude that $\mathbb F/\{0\}$ is a subfield isomorphic to $\mathbb Q$.
Define an embedding $f: \mathbb{Z} \to F$ by $f(n) = n\cdot 1$. This notation means $n\cdot 1 = 1 + ... + 1$, $n$ times if $n>0$ and $n\cdot 1 = - (1+...+1)$ if $n<0$ and $0\cdot 1 = 0$. Therefore, $\mathbb{Z}$ is embedded in $F$, however if $F$ contains by an embedding $\mathbb{Z}$ it must contain by an embedding a field of quotients of $\mathbb{Z}$ i.e. $\mathbb{Q}$.