From the text: In order to determine whether there exists a tensor decomposition of the natural module for a matrix group \(G\) over a field \(K\) it proved to be useful to decide whether or not there exists a tensor decomposition of the characteristic polynomial of \(g\in G\) [C. R. Leedham-Green and E. A. O’Brien, Int. J. Algebra Comput. 7, 541-559 (1997; Zbl 0907.20025)]. This latter problem was the motivation for the present work.
Let \(h\) be a univariate polynomial of degree \(d\) over an algebraically closed field \(K\). If \(d=m+n\) then clearly \(h\) is the product of two polynomials over \(K\) of degrees \(m\) and \(n\). But if \(d=mn\), with \(m,n>1\), then \(h\) is the tensor product (as defined below) of two polynomials, one of degree \(m\) and the other of degree \(n\), if and only if the coefficients \(c_1,\dots, c_d\) of \(h\) define an element \((c_1,\dots, c_d)\) in some \((m+n-1)\)-dimensional variety \(V \subset K^d\). This variety is determined by a prime ideal \(I_{mn}\) in the ring \(K[c_1, \dots, c_d]\). The ideal \(I_{22}\) is easily computed by hand and the ideal \(I_{32}\) is just within the range of machine computation.
Using Gröbner basis algorithms in MAGMA we find necessary and sufficient conditions for a polynomial of degree 6 over any field to be the tensor product of two polynomials, one of degree 3 and one of degree 2.