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.
- 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