Some restricted forms of (D) are weaker than the Axiom of Choice. Fix a field $F$ and consider the stronger statement:
For every $F$-vector space $V$ and every nonzero $v_0 \in V$ there is a $F$-linear functional $f:V\to F$ such that $f(v_0) = 1$.
When $F$ is a finite field, this is a consequence of the Ultrafilter Theorem or, equivalently, the Compactness Theorem for propositional logic.
To see this, consider the following propositional theory with one propositional variable $P(v,x)$ for each pair $v \in V$ and $x \in F$. The idea of the theory is that $P(v,x)$ should be true if and only if $f(v) = x$. The axioms for the theory are:
- $\lnot(P(v,x) \land P(v,y))$ for all $v \in V$ and distinct $x, y \in F$
- $\bigvee_{x \in F} P(v,x)$ all $v \in V$
- $P(v,x) \land P(w,y) \rightarrow P(v + w, x + y)$ for all $v, w \in V$ and $x, y \in F$
- $P(v,x) \rightarrow P(yv,yx)$ for all $v \in V$ and $x, y \in F$
- $P(v_0,1)$
Axiom schemes 1 & 2 ensure that the $P(v,x)$ describe the graph of a function $f:V \to F$. Axiom schemes 3 & 4 ensure that the function $f$ thus described is $F$-linear. Finally, the last axiom 5 ensures that $f(v_0) = 1$. It is clear that every finite subset of the axioms is satisfiable, therefore, by the Compactness Theorem, the whole theory is satisfiable.
