Geodesic
Linear forms and axioms of choice
Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 421-431.

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We work in set-theory without choice ZF. Given a commutative field $\mathbb K$, we consider the statement $\mathbf D (\mathbb K)$: “On every non null $\mathbb K$-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to $\mathbf D (\mathbb K)$ in ZF. Denoting by $\mathbb Z_2$ the two-element field, we deduce that $\mathbf D (\mathbb Z_2)$ implies the axiom of choice for pairs. We also deduce that $\mathbf D (\mathbb Q)$ implies the axiom of choice for linearly ordered sets isomorphic with $\mathbb Z$.
Classification : 03E25, 15A03
Keywords: Axiom of Choice; axiom of finite choice; bases in a vector space; linear forms 
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Morillon, Marianne. Linear forms and axioms of choice. Commentationes Mathematicae Universitatis Carolinae, Tome 50 (2009) no. 3, pp. 421-431. http://geodesic.mathdoc.fr/item/CMUC_2009__50_3_a8/