202 CHAPTER 9. SOME ITEMS WHICH RESEMBLE LINEAR ALGEBRA

for j ≥ p have a factor of p!. If you have

g(x,β 1, · · · ,β m)≡ vpv(m+1)p ((x−β 1)(x−β 2) · · ·(x−β m))p xp−1,

it is symmetric in the β i so all derivatives with respect to x are also symmetric in these β iby Lemma 9.2.2. By the same lemma, for j ≥ p

m

∑i=1

d j

dx j

(g(·,β 1, · · · ,β m)(β i)

1(p−1)!

)=

m

∑i=1

f ( j) (β i)

is symmetric in the β 1, · · · ,β m. Thanks to the factor vpv(m+1)p and the factor p! comingfrom j ≥ p, it is a symmetric polynomial in the vβ i with integer coefficients, each multi-plied by p with the β i roots of Q(x) = vxm + · · ·+ u. By Theorem 9.1.6 this is an integer.As noted earlier, it equals 0 unless j ≥ p when it contains a factor of p. Thus the sum ofthese integers is also an integer times p. It follows that

m

∑i=1

n

∑j=0

f ( j) (β i) = m2 (p) p, m2 (p) an integer. ■

Note that no use was made of p being a large prime number. This will come next.

Lemma 9.2.4 If K and c are nonzero integers, and

β 1, · · · ,β m

are the roots of a single polynomial with integer coefficients,

Q(x) = vxm + · · ·+u

where v,u ̸= 0, then,K + c

(eβ 1 + · · ·+ eβ m

)̸= 0.

Letting

f (x)≡ v(m+1)pQp (x)xp−1

(p−1)!

and I (s) be defined in terms of f (x) as above,

I (s)≡∫ s

0es−x f (x)dx = es

deg( f )

∑j=0

f ( j) (0)−deg( f )

∑j=0

f ( j) (s) ,

it follows,

limp→∞

m

∑i=1

I (β i) = 0 (9.10)

and for n the degree of f (x) ,n = pm+ p− 1, where mi (p) is some integer for p a largeprime number.