Homomorphisms into an Algebraically Closed Field

[Spartan Math]

Okay, so I'm trying to finish of a problem on integral closure and I am rather unsure if the following fact is true:

If L embeds into an algebraically closed field K and F is an algebraic extension of L, then it is possible to extend the embedding of L to F into K.

Now the case where F is a finite extension of L is true, but not quite so sure about the infinite case.

Thoughts would be appreciated.

 

[morphism]

Now the case where F is a finite extension of L is true, but not quite so sure about the infinite case.

The infinite case follows from the finite one by an application of Zorn's lemma.

 

[Spartan Math]

and how exactly does one do that?

 

[Hurkyl]

If you're not a Zorn's Lemma type, then maybe you're a transfinite induction type? Adjoin elements of F to L one at a time, and construct F --> K one bit at a time. Mutter something about well-orderings so that this makes sense.

 

[Spartan Math]

Well, I'm very much a Zorn's Lemma type, if you will. I just wasn't exactly sure how to go about using it.

 

[Hurkyl]

Well, you want to prove the existence of a map from an algebraic extension of L to K, and you already know particular instances. So most naïvely, it seems you'd want the objects of your poset to be such maps.

Then, you'd need an ordering relation to say when one object L-->E-->K is "bigger" than another object L-->F-->K.