Embedding problem for two algebraically closed fields

Question

Give two algebraically closed fields with same characteristic. Can we always embed one of them to the other?

I will appreciate any references.

Answer 1

Yes. If $ L $ and $ M $ are algebraically closed fields of the same characteristic, then let $ K $ be their common prime field; and pick transcendence bases $ S_L, S_M $ respectively. Without loss of generality, assume $ |S_L| \geq |S_M| $. Then, there is an injective map of sets $ S_M \to S_L $, which extends to an embedding $ K(S_M) \to K(S_L) \subset L $ of fields. $ M/K(S_M) $ is an algebraic extension and $ L $ is algebraically closed, thus the embedding $ K(S_M) \to L $ extends to an embedding $ M \to L $.