Exercise 19, § 4

Let C be a principal ideal domain. Consider a system of m « linear congruences »

where the coefficients, the right hand sides, and the element or all belong to C, and where a is nonzero; let A denote the matrix $(\alpha_{ij})$, let B denote the matrix obtained by extending A by the column ($\beta_i$), and let $\delta_k$ (resp. $\delta'_k$) be a gcd of the k-th order minors of A (resp. B). Then the system admits at least one solution in $\mathbf{C}^n$ if and only if: a) If $m \leq n$, a gcd of $a^m, a^{m-1}\delta_1, \ldots, \alpha\delta_{m-1}, \delta_m$ is equal to a gcd of $\alpha^m, a^{m-1}\delta'_1, \ldots, \alpha\delta'_{m-1}, \delta'_m$; b) if $m > n$, a gcd of $\alpha^{n+1}, \alpha^n\delta_1, \ldots, \alpha\delta_n$ is equal-to a gcd of $\alpha^{n+1}, \alpha^n\delta'_1, \ldots, \delta'_{n+1}$. (Reduce the system of linear congruences to a system of linear equations, and use Ex. 18.)