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Let $k$ be a field. In 1984 Andreas Blass proved that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question

Does the axiom "every vector space over $k$ has a basis" imply the axiom of choice ?

What's the current status of the question ? Has there been progress ?

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This has come up before: mathoverflow.net/questions/64219 –  Bruce Westbury Jun 27 '11 at 18:22
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@Bruce: The two questions are clearly different. –  François G. Dorais Jun 27 '11 at 18:31
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Right -- Andreas even says toward the end of the note that the question is open for specific fields like the rationals. I hope that anyone else who votes to close this question leaves a note as to why it should be closed. –  Todd Trimble Jun 27 '11 at 18:53
    
I stand corrected. –  Bruce Westbury Jun 27 '11 at 19:44
    
@Asaf: Thanks. Perhaps you can add references and post it as answer. –  Ralph Jun 27 '11 at 19:55
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up vote 12 down vote accepted

It has been shown for $K=\mathbb F_2$ (the field with two elements) by Keremedis (Available here)

In the dictionary of AC equivalences it shows that not a lot is known on the connection between the existence of a basis over a fixed field and the axiom of choice.

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As far as I can see, Keremedis shows that AC is equivalent to “every generating set of a vector space over $\mathbb F_2$ contains a basis”, which is a stronger statement than “every vector space over $\mathbb F_2$ has a basis”. –  Emil Jeřábek Aug 30 '11 at 15:12
    
Do you mean 'weaker'? –  Jan Veselý Apr 10 '12 at 15:03
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