[FOM] Existence of algebraic closures of fields

[Joachim Reineke]

With the help of galoistheory you only need the axiom of replacement.

Joachim Reineke

----- Original Message ----- 
From: "Aatu Koskensilta" <Aatu.Koskensilta at uta.fi>
To: "Foundations of Mathematics" <fom at cs.nyu.edu>
Sent: Sunday, February 28, 2010 9:42 PM
Subject: Re: [FOM] Existence of algebraic closures of fields


Quoting Colin McLarty <colin.mclarty at case.edu>:

> Does the theorem that every field has an algebraic closure require the
> full strength of the axiom of choice?

No, the theorem is strictly weaker than choice. The compactness
theorem for first-order logic, which Henkin proved equivalent to the
prime ideal theorem, is all we need.

-- 
Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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