Axiom of choice and bases of vector spaces over a fixed field - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:17:34Z http://mathoverflow.net/feeds/question/68950 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68950/axiom-of-choice-and-bases-of-vector-spaces-over-a-fixed-field Axiom of choice and bases of vector spaces over a fixed field Ralph 2011-06-27T18:11:02Z 2011-06-27T20:23:59Z <p>Let \$k\$ be a field. In 1984 Andreas Blass <a href="http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf" rel="nofollow">proved</a> 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 </p> <blockquote> <p>Does the axiom "every vector space over \$k\$ has a basis" imply the axiom of choice ?</p> </blockquote> <p>What's the current status of the question ? Has there been progress ? </p> http://mathoverflow.net/questions/68950/axiom-of-choice-and-bases-of-vector-spaces-over-a-fixed-field/68966#68966 Answer by Asaf Karagila for Axiom of choice and bases of vector spaces over a fixed field Asaf Karagila 2011-06-27T20:23:59Z 2011-06-27T20:23:59Z <p>It has been shown for \$K=\mathbb F_2\$ (the field with two elements) by Keremedis (<a href="http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03305-9/S0002-9939-96-03305-9.pdf" rel="nofollow">Available here</a>)</p> <p>In <a href="http://consequences.emich.edu/file-source/htdocs/conseq.htm" rel="nofollow">the dictionary of AC equivalences</a> 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.</p>