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 $$F$$, an algebraic extension field $$E$$ of $$F$$ and an isomorphism $$\phi$$ mapping $$F$$ onto a field $$F'$$ then $$\phi$$ can be extended to an isomorphism $$\tau$$ mapping $$E$$ onto an algebraic extension $$E'$$ of $$F'$$ (a subfield of the algebraic closure of $$F'$$).

The proof of the isomorphism extension theorem depends on Zorn's lemma.