A system over several fields
Task number: 2481
Solve the following system of equations over \(\mathbb Z_5, \mathbb Z_7\) and \(\mathbb R\).
\( \begin{array}{rlrlrlr} x_1 &+& 2x_2 &+& 4x_3 &=\ & 3 \\ 3 x_1 &+& x_2 &+& 2x_3 &=\ & 4 \\ 2 x_1 &+& 4x_2 &+& x_3 &=\ & 3 \\ \end{array} \)
Resolution
Over \(\mathbb Z_5\): \( \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 3 & 1 & 2 & 4 \\ 2 & 4 & 1 & 3 \\ \end{array}\right) \sim \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 0 & 0 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ \end{array}\right) \) hence \(\mathbf x=(2{,}0,4)^T+(3{,}1,0)^Tp\).
Over \(\mathbb Z_7\): \( \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 3 & 1 & 2 & 4 \\ 2 & 4 & 1 & 3 \\ \end{array}\right) \sim \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 0 & 2 & 4 & 2 \\ 0 & 0 & 0 & 4 \\ \end{array}\right) \), i.e. the system has no solution.
Over \(\mathbb R\): \( \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 3 & 1 & 2 & 4 \\ 2 & 4 & 1 & 3 \\ \end{array}\right) \sim \left(\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 7 & 3 \\ \end{array}\right) \), hence \(\mathbf x=(1{,}1/7{,}3/7)^T\).
