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 $.