Expressing the sum of squares of the roots of a quartic polynomial as a polynomial in its coefficients.

Question

Let $F(x)=X^4+aX^3+bX^2+cX+d$ with $a, b, c, d \in \mathbb{C}$ and let $\alpha_1 , \alpha_2 , \alpha_3 , \alpha_4$ be the roots of $F(x)$. Find a polynomial $G \in \mathbb{Q}[X_1, X_2, X_3, X_4]$ such that $$\alpha_1^2+ \alpha_2^2+\alpha_3^2+\alpha_4^2= G(a, b, c, d).$$

My thoughts: I know that for a given polynomial of $n$ variables in the form of $$P=\sum_{0\leq k \leq n} a_kX^k$$ we have that the coefficient $a_{n-k}=(-1)^ks_k(\alpha_1,..., \alpha_n)$ where $s_k(\alpha_1,..., \alpha_n)$ is the elementary symmetric polynomial of degree $k$.

I'm just not sure how to relate this to the expression above, and any hints would be greatly appreciated.

Answer 1

HINT: $$s_4(\alpha_1,\alpha_2,\alpha_3,\alpha_4)^2-2s_3(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=\cdots$$