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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.

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    $\begingroup$ I think the answer to all three questions is no, by a spreading out argument. $\endgroup$
    – Peter McNamara
    Commented Oct 30, 2018 at 3:56
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    $\begingroup$ The answer to Q2 and Q3 is no, by considering possibilities of the characteristic set. $\endgroup$
    – LeechLattice
    Commented Oct 30, 2018 at 4:34
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    $\begingroup$ Since linear representability of a given finite matroid over a field $F$ is a property of $F$ expressible in first-order logic, negative answers to Questions 2 and 3 follow from the compactness theorem of first-order logic. Then the completeness of the theory of algebraically closed fields of any fixed characteristic implies that a finite matroid representable over a field $F$ (and thus over its algebraic closure) is also representable over the algebraic closure of the prime field of the same characteristic. A negative answer to Q1 then follows. $\endgroup$
    – Andreas Blass
    Commented Oct 30, 2018 at 15:44

1 Answer 1

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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.

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