How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials?
The link I've given to the theorem uses elaborate wordings using 'rings', 'isomorphic', etc.
I completely understand those objects or describings are needed to have a deep understanding, but could anyone try, if it is possible, to explain simply how I could use the theorem to, e.g., factor
$(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2 = a^2b^2(a^2+b^2-2ab)$
without understanding what rings are? I only wish to be able to practically use it.
Best Answer
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OPs symmetric polynomial $P(a,b)$ is
\begin{align*} P(a,b)&=(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2\\ &=a^4b^2-2a^3b^3+a^2b^4\\ &=a^2b^2(a^2-2ab+b^2) \end{align*}
We consider the elementary symmetric polynomials in $2$ variables $a,b$: \begin{align*} e_1&=e_1(a,b)=a+b\\ e_2&=e_2(a,b)=ab \end{align*} We observe, that a factor of $P(a,b)$ is already given as symmetric polynomial \begin{align*} P(a,b)=e_2(a,b)^2\cdot(a^2-2ab+b^2) \end{align*} and we put the focus on \begin{align*} f(a,b)=a^2-2ab+b^2 \end{align*}
Since the polynomial $-4$ is already simple enough, we can calculate $f(a,b)$ as
\begin{align*} f(a,b)=-4e_2(a,b)+g(a,b)=-4e_2(a,b)+e_1(a,b)^2 \end{align*}
We consider the elementary symmetric polynomials in $3$ variables $a,b,c$: \begin{align*} e_1&=e_1(a,b,c)=a+b+c\\ e_2&=e_2(a,b,c)=ab+ac+bc\\ e_3&=e_3(a,b,c)=abc\\ \end{align*}
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We observe according to (2)
\begin{align*} f_2(a,b)=-3e_1(a,b)e_2(a,b)+g_2(a,b)=-3e_1(a,b)e_2(a,b)+e_1(a,b)^3\tag{3} \end{align*} We have systematically found a representation of $q_1(a,b)=f_2(a,b)$ as polynomial of elementary symmetric polynomials and go on with step 1.