In , the choice axiom is equivalent to the statement that for every vector space there exists basis. So more concretely, consider the question of existence of a Hamel bases of the real numbers over the rational numbers.
The choice axiom is equivalent to saying that for any vector space, there is a set of linearly independent vectors that span it. More concretely, we are looking for a Hamel basis for the real numbers in terms of a basis over the rational numbers.