Bases of complex vector spaces and the axiom of choice

Question

In Zermelo-Fraenkel set theory $ZF$ consider the following statement defined for every field $K$:

$B_K$ : Every vector space over $K$ has a basis.

It is well-known that $AC \Rightarrow \forall K (B_K)$. A. Blass proved the converse. In his paper from 1984 he said that for a fixed field $K$, the problem of proving $B_K \Rightarrow AC$ is open. What is the current status? Specifically, I would like to know:

Does $B_{\mathbb{Q}}$ imply $AC$?

Does $B_{\mathbb{C}}$ imply $AC$?

Answer 1

In their paper, Howard and Tachtsis discuss these questions. The paper was published just last year, so I suspect that there hasn't been any significant progress since then.

Paul Howard and Eleftherios Tachtsis, On vector spaces over specific fields without choice, MLQ Math. Log. Q. 59 (2013), no. 3, 128--146.

In general, it seems that the axiom of regularity might play an important part in such proof. Which means that there might be more than just the structure of the vector spaces involved (although this might be mitigated by going at it as Blass did, by showing that you can prove $\sf MC$ rather than $\sf AC$).

If we slightly extend the result to "Every spanning set includes a basis", which in particular means that every vector space has a basis, then Keremedis showed in his paper

Kyriakos Keremedis, Extending independent sets to bases and the axiom of choice, Math. Logic Quart. 44 (1998), no. 1, 92--98.

That over $\Bbb Q$ if every generating set includes a basis, then the axiom of choice holds. I don't know the proof, but it might be possible to extend it to $\Bbb R$ or $\Bbb C$.

All in all, it seems that the questions you particularly address are still very much open, and perhaps new techniques are needed before we can find answers.