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.



Representability of matroids over finite fields
The main results of Rado's [Note on Independence Functions](https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s3-7.1.300) settle all three questions. The first few lines of [Effective Versions of Two Theorems of Rado](http://homepages.ecs.vuw.ac.nz/%7Emayhew/Publications/BFKM.pdf) 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.
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> 
> 


