Axiom of choice and bases of vector spaces over a fixed field - MathOverflow

Axiom of choice and bases of vector spaces over a fixed field

2011-06-27T18:11:02Z

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 ?

Answer by Asaf Karagila for Axiom of choice and bases of vector spaces over a fixed field

2011-06-27T20:23:59Z

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.