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Axiom of Choice; extension of linear forms; non-Archimedean fields; Ingleton's theorem
In set theory without the Axiom of Choice ZF, we prove that for every commutative field $\mathbb K$, the following statement $\mathbf D_{\mathbb K}$: ``On every non null $\mathbb K$-vector space, there exists a non null linear form'' implies the existence of a ``$\mathbb K$-linear extender'' on every vector subspace of a $\mathbb K$-vector space. This solves a question raised in Morillon M., {Linear forms and axioms of choice}, Comment. Math. Univ. Carolin. {50} (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically complete ultrametric valued fields, and show that Ingleton's statement is equivalent to the existence of ``isometric linear extenders''.
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