Show that for any two fields of the same characteristic there is a field into which both of them are embeddable.
So, could you tell me something about this question? I think I have some defect in field theory.
For instance, if these fields have the same char, isn't their alg closure the same?
The basic result here is the following: for characteristic $p \geq 0$ and any uncountable cardinal number $\kappa$, there is an algebraically closed field of characteristic $p$ and cardinality $\kappa$, and any two such fields are isomorphic. A proof is given in $\S$ 12.2 of these notes. (This is not true for $\kappa = \aleph_0$, i.e., for countably infinite fields. The "real answer" is that algebraically closed fields are determined by their characteristic and their transcendence degree over the prime subfield, but when the cardinality is uncountable, it must be equal to the transcendence degree over the prime subfield.)
The result you want follows rather easily from this. (In fact, the very next result in the notes linked to above is essentially the assertion you're asking about, and the proof is left as an exercise.)