According to Wikipedia 
<https://en.wikipedia.org/wiki/Algebraic_matroid>
"For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide"


I cannot find the two references it provides. There is a theorem that if you are algebraic over F then you are linear over an extension of F. In the case of characteristic zero, or more precisely, the complex numbers, is it true without going to an extension?


I checked the examples at the end of Oxley but none seem to have this property.


EDIT: According to the review in Mathscinet, this is proved in
Ingleton, A. W. Representation of matroids. 1971 Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) pp. 149–167 Academic Press, London 


but I cannot access that reference, I just want to double check, or maybe see if there is another reference.



Over the complex numbers, is there an example of an algebraic but non-representable matroid?
Algebraic over $\mathbb{C}$ implies realizable over $\mathbb{C}$. If your matroid is algebraic over $\mathbb{C}$, then it is realizable over some $\mathbb{C} \subset K$, so it is realizable over the algebraic closure $\overline{K}$, which explicitly means that a list of equalities and inequalities over $\overline{K}$ has a solution. But then these equations and inequalities have a solution over $\mathbb{C}$ as well.


