[FOM] Existence of algebraic closures of fields

Joachim Reineke reineke at math.uni-hannover.de
Sun Feb 28 17:02:10 EST 2010

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

FOM mailing list
FOM at cs.nyu.edu

More information about the FOM mailing list