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

The proof of the isomorphism extension theorem in its most general setting, i.e. for the case of a field extension of infinite degree, depends on Zorn's lemma.

References

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