It is known that "every vector space has a basis" $\Leftrightarrow$ Axiom of choice
but do we have the same thing with a vector space over $\mathbb{Q}$?
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As a small note, you might be interested in learning that for any field $k$, "every vector space over a field extension of $k$ has a basis" $\implies$ AC. – Mees de Vries Apr 06 '17 at 22:44
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Do you know a demonstration of what you claim (or maybe a link for the demonstration) ? – ketherok Apr 08 '17 at 16:02
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I'm sorry, I don't. It was presented to me in a lecture. As far as I know, it follows immediately from the standard proof that the existence of bases implies AC. – Mees de Vries Apr 08 '17 at 16:54