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Qiaochu Yuan
Qiaochu Yuan
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The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, which assertsasserting that it can be proven usingthe proof only requires the ultrafilter lemma, which is known to be strictly weaker. Questions:

  • Is it known whether the two are equivalent in ZF?
  • Would anyone like to give a quick sketch of the construction assuming the ultrafilter lemma? I dislike the usual construction and am looking for others.

The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, which asserts that it can be proven using the ultrafilter lemma, which is known to be strictly weaker. Questions:

  • Is it known whether the two are equivalent in ZF?
  • Would anyone like to give a quick sketch of the construction assuming the ultrafilter lemma? I dislike the usual construction and am looking for others.

The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, asserting that the proof only requires the ultrafilter lemma. Questions:

  • Is it known whether the two are equivalent in ZF?
  • Would anyone like to give a quick sketch of the construction assuming the ultrafilter lemma? I dislike the usual construction and am looking for others.
Source Link
Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

Is the statement that every field has an algebraic closure known to be equivalent to the ultrafilter lemma?

The existence and uniqueness of algebraic closures is generally proven using Zorn's lemma. A quick Google search leads to a 1992 paper of Banaschewski, which I don't have access to, which asserts that it can be proven using the ultrafilter lemma, which is known to be strictly weaker. Questions:

  • Is it known whether the two are equivalent in ZF?
  • Would anyone like to give a quick sketch of the construction assuming the ultrafilter lemma? I dislike the usual construction and am looking for others.