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

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.

• – lhf
Sep 27 '20 at 10:59

## 1 Answer

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