Isomorphism extension theorem

Short description: Theorem in field theory

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 ϕ mapping F onto a field F′ then ϕ can be extended to an isomorphism τ mapping E onto an algebraic extension E′ of F′ (a subfield of the algebraic closure of 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.

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