Isomorphism extension theorem

In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to a larger field.
Isomorphism extension theorem.
The theorem states that given any field formula_1, an algebraic extension field formula_2 of formula_1 and an isomorphism formula_4 mapping formula_1 onto a field formula_6 then formula_4 can be extended to an isomorphism formula_8 mapping formula_2 onto an algebraic extension formula_10 of formula_6 (a subfield of the algebraic closure of formula_6).
The proof of the isomorphism extension theorem depends on Zorn's lemma.