Exercise 20, § 2

Algebra, VI, p. 42

Let K be a maximal ordered field and

f(T) = a_0 + \binom{n}{1} a_1 T + \binom{n}{2} a_2 T^2 + \cdots + a_n T^n

a polynomial in

K[T]

. For all integers

p, q

such that

0 \leq p < p + q \leq n

, the number of roots of the polynomial

a_p + \binom{q}{1} a_{p+1} T + \binom{q}{2} a_{p-2} T^2 + \cdots + a_{p+q} T^q

counted with multiplicity, which do not belong to K is at most equal to the number of roots of

f

, counted with multiplicity, which do not belong to K (use Ex. 12). Deduce that if

b_1, \ldots, b_n

are

n

distinct elements

> 0

in K, and if

(T + b_1) \cdots (T + b_n) = T^n + \binom{n}{1} m_1 T^{n-1} + \cdots + m_n

then

m_k^{1/k} > m_{k+1}^{1/(k+1)} \quad \text{for} \quad 1 \leq k \leq n-1 .

Solution

No solution has been written yet.