I would like some help in proving the following statement:
A field $K$ has characteristic 0 if and only if $\mathbb{Q}$ is a subfield of $K$.
So, the way I have approached this is by first assuming that $K$ has characteristic 0 and then we know that there is an embedding of $\mathbb{Z}$ into K and since $n.1$ is in K, then, their inverses will also be in $K$ and hence the subfield that we get is the one generated by ${1}$ and this is the prime subfield of K and this is isomorphic to $\mathbb{Q}$. So $K$ contains $\mathbb{Q}$. But I am not sure how to progress the other way.
(If you feel that the reasoning I have used above is wrong/can be improved, I would be extremely happy for it to be corrected.)
Best Answer
Your proof is fine. Actually, you have done the hard direction the other direction is easier ! Suppose that $\mathbb{Q}$ is a subfield of $K$. Then if we had an equation of the form $$ 1 + 1 + \ldots + 1 = 0 $$ in $K$, we would also have it in $\mathbb{Q}$ (because the $1$ is the same whether we view it in $K$ or in the subfield $\mathbb{Q}$). But we have no such equation in $\mathbb{Q}$.