I'm trying to generalize Birkhoff's theorem and prove that over any field $\mathbb{F}$ any matrix whose sum of rows and columns is equal to some constant, is a linear combination of permutation matrices.
For fields where the characteristic $p$ does not divide $n$, we can use a dimensional argument: compare the dimension over $\mathbb{F}$ of the span of all permutation matrices on $n$ points with the algebra $Mat_{n-1}(\mathbb{F})\oplus \mathbb{F}$. These algebras are isomorphic as it is sufficient for $p$ to not be a factor of $n$ so that it the representation decomposes into the direct sum of the trivial and the permutation representation. This does not work otherwise. Any hints?