Let $k$ be a field. In 1984 Andreas Blass [proved](http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf) 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 



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> Does the axiom "every vector space over $k$ has a basis" imply the axiom of choice ?
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What's the current status of the question ? Has there been progress ? 



Axiom of choice and bases of vector spaces over a fixed field 
It has been shown for $K=\mathbb F\_2$ (the field with two elements) by Keremedis ([Available here](http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03305-9/S0002-9939-96-03305-9.pdf))


In [the dictionary of AC equivalences](http://consequences.emich.edu/file-source/htdocs/conseq.htm) 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.


