Categoricity for ACFp
Theorem: (Steinitz , Crelles J., 1910) If K and L are algebraically closed fields of the same characteristic and of the same transcendence degree over their common prime field F, then K ≅ L.
Proof Sketch:
  • Let X ⊂ K and Y ⊂ L be transcendence bases. By hypothesis there is a bijection f:X → Y.
  • The bijection f induces an isomorphism g:F(X) ≅ F(Y) between the field generated by X and the field generated by Y.
  • In general, any isomorphism between fields extends to an isomorphism of their algebraic closures.
  • By maximality of X and Y, the algebraic closure of F(X) is K and the algebraic closure of F(Y) is L