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 , an algebraic extension field  of  and an isomorphism  mapping  onto a field  then  can be extended to an isomorphism  mapping  onto an algebraic extension  of  (a subfield of the algebraic closure of ).

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

References
 D.J. Lewis, Introduction to algebra, Harper & Row, 1965, Chap.IV.12, p.193.

Field (mathematics)
Theorems in abstract algebra