I was wondering if the following lemma was easy to prove. I got a little tripped up when I saw the explicit condition that the matrix could have elements in any field not necessarily finite, so I didn't know if a standard proof from linear algebra would still apply.
Let $M$ be a matrix with entries in a (possibly infinite) field $F$.
Suppose that there exists a minor $m_n$ of order $n$ for $M$ that is nonzero and such that all minors of order $n+1$ which contain $m_n$ are zero.
How do you show the rank of the matrix is $n$?