Consider some numbers like 1, 2, -5,\( \sqrt{2} \), \( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \). Can we write them as the solution to an equation \( f(x)=0 \)? Yes, of course! We can write simple equations such as \( x-1=0 \), \( x-2=0 \), \( x+5=0 \), \( x-\sqrt{2}=0 \), and \( x+\frac{1}{2}-i\frac{\sqrt{3}}{2}=0 \). But what about if we add a restriction on the equation f(x) that the numbers which appear should be rational.

\[\sigma = \begin{pmatrix} x_1 & x_2 & x_3 & \cdots & x_n \\ \sigma(x_1) &\sigma(x_2) & \sigma(x_3) & \cdots& \sigma(x_n)\end{pmatrix}.\] \[\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix};\] \[\sigma=\begin{pmatrix} 3 & 2 & 5 & 1 & 4 \\ 4 & 5 & 1 & 2 & 3\end{pmatrix}.\] \[\sigma = (125)(34).\] \[\sigma=\begin{pmatrix} 3 & 2 & 5 & 1 & 3 & 2 & 5 & 1 & 3 & 2 & 5 & 1 & 3 & 2 & 5 & 1 & 4 \\ 4 & 5 & 1 & 2 & 3 & 2 & 5 & 1 & 3 & 2 & 5 & 1 & 3 & 2 & 5 & 1 & 4\end{pmatrix}.\]