## Marianne Morillon

*Linear forms and axioms of choice*

Comment.Math.Univ.Carolinae 50,3 (2009) 421-431.**Abstract:**We work in set-theory without choice {\bf 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 {\bf 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 $\mathbb Z$.

**Keywords:** Axiom of Choice, axiom of finite choice, bases in a vector space, linear forms

**AMS Subject Classification:** 03E25 15A03

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