Algebraically closed field

Embedding an algebraic extension into an algebraically closed field

Assume Zorn’s lemma. If is a field extension with algebraically closed and is any algebraic extension, then there exists a morphism of field extensions

Proof

Consider the restricted Poset of intermediate extensions

where we note all fields in are algebraic extensions of . We show that every ^chain in has an upper bound. Let which is a field. If , then we define for some , which is clearly independent of the choice of . Then is an upper bound of .

By Zorn’s lemma, has a maximal element . Since is algebraic,

We claim that , whence is the desired morphism.

Suppose towards contradiction there exists and consider the simple extension . Since is algebraic over , it is algebraic over , thus it is the root of an irreducible . Abusing notation to invoke the induced homomorphism

let , which is irreducible over , and has a root in — here we use that is algebraically closed. Now by ^P2, the isomorphism lifts to an isomorphism

sending to , contradicting the maximality of . Therefore , and we’re done.

This lemma is used to prove uniqueness of the Algebraic closure.

---

tidy | en | sembr