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.