Abstract:In set theory without the Axiom of Choice {\bf 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., {\it Linear forms and axioms of choice\/}, Comment. Math. Univ. Carolin. {\bf 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''.
Keywords: Axiom of Choice; extension of linear forms; non-Archimedean fields; Ingleton's theorem
DOI: DOI 10.14712/1213-7243.2015.223
AMS Subject Classification: 03E25 46S10