Exercise 12, § 8

Let K be a field, $v$ a valuation on K, A the ring of the valuation $v$ and $\Gamma$ its order group.

(a) Let $f(X) = a_0 X^n + a_1 X^{n-1} + \ldots + a_n$ be a polynomial in $A[X]$ such that $v(a_0) = 0$, all of whose roots belong to K; let $x_i$ ($1 \leq i \leq r$) be these distinct roots and let $k_i$ be the order of multiplicity of $x_i$ ($1 \leq i \leq r$). Let

another polynomial in $\mathbf{A}[X]$ such that $v(b_0) = 0$ and the $y_i$ belong to K. Show that there exists $\lambda_0 \in \Gamma$ such that $v(x_i - x_j) < \lambda_0$ for every ordered pair of distinct indices $i, j$ and, for all $A \geq A_n$, there exists $\mu \geq \lambda$ such that the relations $v(a_i - b_i) \geq \mu$ for all $i$ imply that, for $1 \leq i \leq r$, there are exactly $k_i$ indices $h$ such that $v(x_i - y_h) \geq A$. (First evaluate $v(f(y_h))$ in two ways to show that necessarily $v(x_i - y_h) \geq \lambda$ for at least one $i$; then evaluate

at suitable points $z \in K$.) If further $b_i = a_i$ except for $i = n - 1$, show that, so long as $\lambda_0$ is sufficiently large, the $y_h$ are all distinct (evaluate $f'(y_h)/f(y_h)$ assuming that $y_h$ is not a simple root of $g$).

(b) Suppose henceforth that K is the algebraic closure of a subfield $K_0$ such that $A_n = A \cap K_0$ is Henselian. Suppose that the polynomial $f$ belongs to $A_0[X]$ and is irreducible and separable over $K_0$. Let $z \in K$ be such that

for all $j \neq i$; then show that $K_0(x_i) \subset K_0(z)$. (Show that $z - x_i$ is equal to all its conjugates with respect to $K_0(x_i)$.)

(c) Suppose that $f \in A_0[X]$ is separable and let $f = a_0 \prod_{i=1}^r f_i$ be the decomposition off into monic irreducible polynomials in $K_0[X]$ (which belong in fact to $A_0[X]$, cf. Chapter V, § 1, no. 3, Proposition 11). Show (in the notation of (a)) that, if $\mu$ is taken sufficiently large, the decomposition of g into monic irreducible factors in $K_0[X]$ can be written as $b_0 \prod_{i=1}^r g_i$, where, for all $i$, $g_i$ is of the same degree as $f_i$ (a decomposition said to be "of the same type" as that of $f$) and further the splitting fields of $f_i$ and $g_i$ are the same for all $i$. (Show first that g is separable; consider then a finite Galois extension of $K_0$ containing the roots of $f$ and g and consider the way in which the Galois group of this extension permutes these roots; finally use (b).)

(d) Give an example where $f$ is irreducible but not separable and where, for all $\mu \geq A_n$ there exists a polynomial

where $v(a_i - b_i) \geq \mu$ for all $i$, which is not irreducible (take $K_0$ such that $K \neq K_0$ and K is a radicial extension of $K_0$).