Let $K$ be a field, and $F_K$ be the fraction field of the polynomial ring $R_K$ in $n^2$ indeterminates $X_{11},X_{12},...,X_{nn}$ over $K$. Now set $A = (X_{ij})_{i,j} \in M_n (F_K)$, and let $\chi_A$ be the characteristic polynomial of $A$.

**Question** : Is it always true that $\chi_A$ is irreducible over $F_K$ ?

Some thoughts :

- Since $R_K$ is a UFD, it is sufficient to check irreducibility in $R_K[X]$.
- If $ |K| \geq n$, then $\chi_A$ is separable (by evaluation of $A$ to a diagonal matrix with distinct diagonal entries).
- If $K[X]$ contains some irreducible $f$ of degree $n$, then this is true (just evaluate $A$ to the companion matrix of $f$ and observe that the evaluation morphism cannot increase degrees). For example, this yields the result when $K = \mathbb{Q}$ or when $K$ is a finite field.
- More generally, a non trivial factor of $\chi_A$ is unlikely to exist, since it would give a generic factorization of the characteristic polynomial of a $n \times n$ matrix with coefficients in $K$.

Even in the case $K=\mathbb{C}$, I cannot prove (or disprove) the irreducibility of $\chi_A$.