Exercise 1, § 6

Algebra, IV, p. 98

Let $r_k$ be the polynomial with integer coefficients such that

p_k = r_k(s_1, \ldots, s_k) .

a) Show that

r_k(Y_1, \ldots, Y_s)

contains the terms

Y_1^k

and

(-1)^{k-1} k Y_k

. b) Show that

r_k(Y_1, \ldots, Y_k) = \sum_{p=0}^{k-1} (-1)^p \left| \begin{array}{cccccc} Y_{p+1} & Y_{p+2} & Y_{p+3} & \ldots & Y_{k-1} & Y_k \\ 1 & Y_1 & Y_2 & \ldots & Y_{k-p-2} & Y_{k-p-1} \\ 0 & 1 & Y_1 & \ldots & Y_{k-p-3} & Y_{k-p-2} \\ 0 & 0 & 0 & \ldots & Y_1 & Y_2 \\ 0 & 0 & 0 & \ldots & 1 & Y_1 \end{array} \right|

Solution

No solution has been written yet.