Exercise 20, § 4

a) Let C be a principal ideal domain, and let

be a p-linear form on $E^p$, where E is the C-module $C^n$ (and where $\xi_{ij}$ denotes the j-th coordinate of $x_i$). Show that if 6 is a gcd of the coefficients $\alpha_{i_1 \ldots i_p}$ then there exists an element $(a_i) \in E^p$ such that $f(a_1, \ldots, a_p) = 6$ (reduce to the case $6 = 1$, and argue by induction on p, using Th. 1 of VII, p. 18). b) Let $n > 1$ be an integer, and $\alpha_i \ (1 \leq i \leq n)$ be n elements of C with gcd 6. Show that there exists a square matrix A of order n over C whose first column is $(\alpha_i)$ and whose determinant is 6 (use a)).