# “Two algebraically closed fields with char $0$ and same cardinality $\kappa> \aleph_0$ are isomorphic.” Any reference for this result?

I already know how to prove the result, however I need a reliable source I can cite so I can omit the proof. I'm having a terrible hard time trying to find it throughout Google. Can anyone please help me?

• It's not true. You need the same characteristic as well as the same cardinality. – Lord Shark the Unknown Dec 12 '17 at 4:50
• sure! my mistake; edited. thanks! – Roland Dec 12 '17 at 4:51
• Note that Proposition 4.4 points towards Lang’s Algebra Chapter $X$, Section $1$. – Rohan Dec 12 '17 at 4:58

If you want just some published statement to cite, this statement is the case $p=0$ of Proposition 2.2.5 in David Marker's Model Theory: An Introduction.
Let $K$ and $L$ be two algebraically closed field of the same characteristic and of the same uncountable cardinality. Write $E$ and $F$ for the algebraic closures of the prime fields of $K$ and $L$, respectively. Of course, $E$ and $F$ are isomorphic. Since $K$ and $L$ have the same cardinality $\kappa>\aleph_0=|E|=|F|$, the transcendence degrees of $K$ over $E$ and $L$ over $F$ are the same. Thus, any transcendence bases $B$ of $K$ over $E$ and $C$ of $L$ over $F$ have the same cardinality $\kappa$.
Now, we pick an arbitrary field isomorphism $\sigma:E\to F$. Extend $\sigma$ to $E(B)$ and $F(C)$, and the resulting extension $\tilde{\sigma}$ is still a field isomorphism. Finally, since $K$ is the algebraic closure of $E(B)$ and $L$ is the algebraic closure of $F(C)$, the isomorphism $\tilde{\sigma}$ extends to a field isomorphism $\bar{\sigma}:K\to L$.