Exercise 22, § 10

Let m be an even integer > 0, r an integer $\geq 3$ and $n_1 < \ldots < n_{r-2}$, r - 2 even integers. Put $g(X) = (X^2 + m)(X - n_1) \ldots (X - n_{r-2})$. a) Put $f(X) = g(X) - 2$. Show that $f(X)$ has at least $r - 2$ real roots. Calculate the sum of the squares of the roots of f in C. Deduce that for $m \geq \sum n_i^2 / 2$, $f(X)$ is a polynomial with integer coefficients having $r - 2$ real roots and two non-real conjugate roots. b) Show that $f(X)$ is irreducible in $\mathbf{Q}[X]$. (Write $f(X) = X^r + \sum_{k=1}^r a_k X^{r-k}$; show that $a_r$ is even and that $a_r$ is not divisible by 4 ; apply Ex. 21.) c) Show that when r is prime and $m \geq \sum n_i^2 / 2$, then the Galois group G of the splitting field of f is isomorphic to $\mathfrak{S}_r$ (cf. V, p. 158, Ex. 17). d) Show that when $r = 4$ and $m \geq \sum n_i^2 / 2$, the degree over Q of the splitting field is equal to 8 or 24, and that it is equal to 8 if and only if $n_1 = -n_2$ or also if and only if $f(X)$ is of the form $P(X^2)$, where P is a polynomial of degree 2. \*