TITLE: Representability of matroids over finite fields
QUESTION [6 upvotes]: I have several questions regarding representability of matroids.
Question 1. Does there exist a finite matroid that is representable over an infinite field, but is not representable over any finite field?
Question 2. Does there exist a finite matroid that is representable over a field of characteristic $0$, but is not representable over any field of positive characteristic?
Question 3. Does there exist a finite matroid that is representable over a field of characteristic $0$, but $\{\mathrm{char}(F): M \text{ is representable over }F\}$ is a finite set?
I care about Question 1 most, and find other quetions also interesting. I briefly checked Oxley, Matroid theory but did not find an answer.

REPLY [7 votes]: The main results of Rado's Note on Independence Functions settle all three questions. The first few lines of Effective Versions of Two Theorems of Rado give a perfect recap of those results, so here they are verbatim:

Our starting point is given by the following two theorems of Rado [5].
Theorem 1 (Rado, 1957). Let $M$ be a matroid representable over a field $K$. Then $M$ is representable over a simple algebraic extension of the prime field of $K$.
Theorem 2 (Rado, 1957). Let $K$ be an extension field of $\mathbb{Q}$ of degree $N$ , and let $M$ be a matroid representable over $K$. Then there is a positive integer $c$ such that given any prime $p > c$ there is a positive integer $k = k(p) ≤ N$ such that $M$ is representable over $GF(p^k)$. For infinitely many $p$, $k(p) = 1$.
Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field.