Exercise 23, § 10

Let k be a field of characteristic 2. a) Let $A_1, \ldots, A_n$ be indeterminates, K the field $k(A_1, ..., A_n)$, E a splitting field of the polynomial $P = X^n + A_1 X^{n-1} + ... + A_n \in K[X]$ and $X_1, ..., X_n$ the roots of P in E, so that $E = k(X_1, ..., X_n)$ (V, p. 21). We identify the Galois group of E/K with the symmetric group $S_n$ acting by permutations of the variables $X_1, ..., X_n$ (V, p. 59). Put

Then $\beta \notin K$ and We have $\sigma(\beta) = \beta$ for $\sigma \in U_n$ and $\sigma(\beta) = \beta + 1$ for $\sigma \in S_n - U_n$. Deduce that $K(\beta)$ is the unique quadratic subextension of $E/K$. b) Let $D \in \mathbf{Z}[T_1, ..., T_n]$ be the discriminant of the polynomial $X^n + T_1 X^{n-1} + ... + T_n$. Show that there exist Q, R, S $\in \mathbf{Z}[T_1, ..., T_n]$ with and For every triple of elements U, V, W of $\mathbf{Z}[T_1, ..., T_n]$ such that $D = U^2 - 4V + 8W$ there exists $M \in \mathbf{Z}[A_1, ..., A_n]$ such that c) Let $f = X^n + a_1 X^{n-1} + ... + a_n$ be a separable polynomial of $k[X]$ and G the Galois group of a splitting extension. Show that $(a_1, ..., a_n)$ is substitutable in C and that the following conditions are equivalent (i) every element of G induces an even permutation of the roots of f ; (ii) there exists $a \in k$ such that $C(a_1, ..., a_n) = a^2 + a$. \* d) Henceforth assume that k is finite and choose the elements U, V, W of $\mathbf{Z}[T_1, ..., T_n]$ arbitrarily to satisfy the condition stated in b). Show that (i) and (ii) of c) are equivalent to the following conditions : (iii) (iv) (v) The number of irreducible factors of f is congruent to n modulo 2 (use Ex. 22 of V, p. 165). \*