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 $\varphi$ mapping $F$ onto a field $F^{\prime }$ then $\varphi$ can be extended to an isomorphism $\tau$ mapping $E$ onto an algebraic extension $E^{\prime }$ of $F^{\prime }$ (a subfield of the algebraic closure of $F^{\prime }$).

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.