When does the ring \(K[y]\) have the coefficient assignment property?
Let \(R\) be a commutative ring with \((A,B)\) an \(n\)-dimensional controllable system over \(R\). Thus, \(A\) is an \(n\times n\) matrix, \(B\) is an \(n\times n\) matrix, and the \(R\)-module generated by the columns of the matrix \([B,AB, \dots A^{n-1} B]\) is \(R^n\). If \(R\) is a field, then the controllability of a system is equivalent to any of the following three conditions: 1. There exists a matrix \(F\) and a vector \(\nu\) such that \(B\nu\) is a cyclic vector for the matrix \(A+BF\). 2. For each monic, \(n\)-th degree polynomial \(f(x)\in R[x]\), there exists a matrix \(F\) such that the characteristic polynomial of \(A+BF =f(x)\). 3. For each collection \(\{r_1, \dots, r_n\}\) of elements of \(R\), there exists a matrix \(F\) such that the characteristic polynomial of \(A+BF =(x-r_1) \cdots (x-r_n)\). Over an arbitrary ring, these are no longer equivalent. Instead, for a system \((A,B)\) over \(R\), we have that \((1) \Rightarrow (2) \Rightarrow (3)\) and that each of these conditions implies the controllability of the system. A ring \(R\) is called a \(CA\)-ring if condition (2) is satisfied for all controllable systems over \(R\). Now, it was shown by \textit{R. Bumby}, \textit{E. D. Sontag}, \textit{H. J. Sussman}, and \textit{W. Vasconcelos} [J. Pure Appl. Algebra 20, 113-127 (1981; Zbl 0455.15009)] that \(\mathbb{R}[y]\) is not a \(CA\)-ring. Our result is the following: Theorem. Let \(K\) be a field with \(y\) an indeterminate over \(K\). Let \(q\) be a prime integer different from the characteristic of \(K\) and suppose that \(K\) contains all \(q\)-th roots of unity. If \(K[y]\) is a \(CA\)-ring, then \(K\) is closed under taking \(q\)-th roots; that is, the map \(\varphi: K\to K\) defined by \(\varphi (x)= x^q\) is surjective. In particular, suppose that \(K\) is a field of characteristic 0 and that for each positive integer \(n\), \(K\) contains all the \(n\)-th roots of unity. If \(K[y]\) is a \(CA\)-ring, then for each positive integer \(n,K\) is closed under taking \(n\)-th roots.
scientific article; zbMATH DE number 937161