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 [math]\displaystyle{ F }[/math], an algebraic extension field [math]\displaystyle{ E }[/math] of [math]\displaystyle{ F }[/math] and an isomorphism [math]\displaystyle{ \phi }[/math] mapping [math]\displaystyle{ F }[/math] onto a field [math]\displaystyle{ F' }[/math] then [math]\displaystyle{ \phi }[/math] can be extended to an isomorphism [math]\displaystyle{ \tau }[/math] mapping [math]\displaystyle{ E }[/math] onto an algebraic extension [math]\displaystyle{ E' }[/math] of [math]\displaystyle{ F' }[/math] (a subfield of the algebraic closure of [math]\displaystyle{ F' }[/math]).

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.