Exercise 13, § 5

Let $(Z_{ij})$ ($1 \leq i, j \leq n$) be $n^2$ indeterminates over a field $K_0$. Let A be the polynomial ring $K_0[Z_{11}, ..., Z_{nn}]$, and let $K = K_0(Z_{ij})$ be its field of fractions ; let $\chi_U$ be the characteristic polynomial of the matrix $U = (Z_{ij})$ over K. a) Show that $\chi_U$ is a separable irreducible polynomial (show that, if $\chi_U$ were reducible over K, it would be the product of two polynomials in $K_0[X, Z_{,,}]$ of degree $> 0$ in X : hence deduce that the characteristic polynomial of every square matrix of order n over K would be reducible ; then derive a contradiction from the example of a polynomial with algebraically independent coefficients over $K_0$; use the analogous method to prove separability). b) Show that $U$ is similar to a matrix in which every entry is equal to 0, 1 or a coefficient of the polynomial $\chi_U$ (cf. VII, p. 30, formula (2)). Deduce that, if $K_0$ is an infinite field, and p is a polynomial in A such that $p(u_{ij}') = p(u_{ij}'')$ whenever the matrices $(u_{ij}')$ and $(u_{ij}'')$ over K, are similar, then p belongs to the subring of A generated by the coefficients of $\chi_U$. State and prove the analogous result for rational functions $r \in K = K_0(Z_{ij})$.