Exercise 16, § 2

Algebra, VI, p. 41

Let K be a maximal ordered field and let

g(T) = a_0 + a_1 T + \cdots + a_n T^n

be a nonzero polynomial in K[T] whose roots all lie in K. Show that for any polynomial $f \in K[T]$ , the number of roots of the polynomial

a_0 f(T) + a_1 f'(T) + a_2 f''(T) + \ldots + a_n f^{(n)}(T)

which do not belong to K, counted with multiplicity, is at most equal to the number of roots of $f$ , counted with multiplicity, which do not belong to K (use Ex. 12 for $n = 1$ , then proceed by induction). Deduce that the polynomial

a_0 + \frac{a_1}{1!} T + \frac{a_2}{2!} T^2 + \cdots + \frac{a_n}{n!} T^n

has all its roots in K.

Solution

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