Steinitz's theorem on algebraically closed fields

Remarks

If the characteristics agree, then $P_K \cong P_L$, and a bijection between transcendence bases $B$ of $K$ and $B'$ of $L$ gives an isomorphism $P_K(B) \cong P_L(B')$. Since $K$ and $L$ are algebraic closures of these fields, the isomorphism extends to $K \cong L$ by uniqueness of the algebraic closure. The converse holds because an isomorphism preserves the characteristic and maps a transcendence basis to a transcendence basis. For an uncountable algebraically closed field, the transcendence degree over the prime field equals the cardinality of the field. So two uncountable algebraically closed fields of the same characteristic and the same cardinality are isomorphic; for example, $\C$ is isomorphic to an algebraic closure of $\Q(x_i : i \in \R)$.

Going deeper

For a student

The theorem shows that the algebraic closure $\overline{\Q}$ and $\C$ are not isomorphic: one has transcendence degree $0$, the other has transcendence degree the continuum. It is the field-theoretic source of the fact that the theory of algebraically closed fields of a fixed characteristic is categorical in each uncountable cardinality. Its proof uses the existence of transcendence bases and the uniqueness of algebraic closures, both of which rest on Zorn's lemma.