# 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 $\displaystyle \mathbb Q$.

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

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 $\displaystyle \mathbb Q$.

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