## Linear forms and axioms of choice.(English)Zbl 1212.03034

Summary: We work in set theory without choice $$\mathbf {ZF}.$$ 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 $$\mathbf {ZF}.$$ 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 $$\mathbb Z$$.

### MSC:

 03E25 Axiom of choice and related propositions 15A03 Vector spaces, linear dependence, rank, lineability
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