# Linear extenders and the Axiom of Choice

## TL;DR

This paper explores the implications of the Axiom of Choice in set theory, establishing conditions under which linear extenders exist in vector spaces over fields, including ultrametric valued fields, and linking these to specific axioms.

## Contribution

It proves that certain statements about linear forms imply the existence of linear extenders in ZF set theory and generalizes results to ultrametric fields, addressing a previously open question.

## Key findings

- D_{ ext{IK}} implies existence of linear extenders in ZF.
- Equivalence of Ingleton's statement and isometric linear extenders in ultrametric fields.
- Extension of results to spherically complete ultrametric valued fields.

## Abstract

In set theory without the axiom of Choice ZF, we prove that for every commutative field IK, the following statement D_{\IK}: "On every non null IK-vector space, there exists a non null linear form" implies the existence of a IK-linear extender on every vector subspace of a $\IK$-vector space. This solves a question raised in \cite{Mo09}. 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".