Exercise 15, § 6

We resume the hypotheses and notations of exercise 12. Let $\Delta_1(X)$ be the largest common divisor of the coefficients of B(X); the polynomial $\Delta_1(X)$ is assumed to be monic.

a) The polynomial $\Delta_1(X)$ divides Δ(X) and $\Delta_1(r)\not = 0$.

b) Deduce that the real roots of $\Delta_1(X)$ are strictly smaller than $r$. Conclude that $\Delta_1(r)>0$. c) Put C$(X) =\frac{1}{\Delta_1(X)}B$(X). This is a matrix with coefficients in $\mathbf{R}[X]$. Now prove that C$(r)\succ 0$.