Rationalize irrational equation

Question

$$ \begin{cases} \sigma_{1,(2)}=x_1+x_2\\ \sigma_{2,(2)}=x_1x_2\\ \end{cases}\\ \color{red}{ \begin{align*} &&p_2&=\sqrt{x_1}+\sqrt{x_2}\\ &\Rightarrow&\left({p_2}^2-\sigma_{1,(2)}\right)^2&=4\sigma_{2,(3)}\\ \end{align*}} $$ $$ \begin{cases} \sigma_{1,(3)}=x_1+x_2+x_3\\ \sigma_{2,(3)}=x_1x_2+x_1x_3+x_2x_3\\ \sigma_{3,(3)}=x_1x_2x_3 \end{cases}\\ \color{red}{ \begin{align*} &&p_3&=\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}\\ &\Rightarrow&\left[\left({p_3}^2-\sigma_{1,(3)}\right)^2-4\sigma_{2,(3)}\right]^2&=64{p_3}^2\sigma_{3,(3)}\\ \end{align*}} $$ What about the situation of more variables?

How to rationalize it $$p_4=\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$

I have ran it above through Wolfram Mathematica and no result return.

Answer 1

\begin{gather*} &256\sigma_{4,(4)}\left(3p^4-2p^2\sigma_{1,(4)}-{\sigma_{1,(4)}}^2+4\sigma_{2,(4)}\right)^2\\ \\ {\large=}\!\!&\left(\left(\left(p^2-\sigma_{1,(4)}\right)^2-4\sigma_{2,(4)}\right)^2-64p^2\sigma_{3,(4)}+64\sigma_{4,(4)}\right)^2\\ \end{gather*}

\begin{cases} \sigma_{1,(4)}=x_1+x_2+x_3+x_4\\ \sigma_{2,(4)}=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4\\ \sigma_{3,(4)}=x_1x_2x_3+x_2x_3x_4+x_1x_3x_4+x_1x_2x_4\\ \sigma_{4,(4)}=x_1x_2x_3x_4\\ \end{cases} \begin{gather*} p=\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4} \end{gather*}