## Marianne MorillonLinear extenders and the Axiom of Choice

Comment.Math.Univ.Carolin. 58,4 (2017) 419-434.

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

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