A note on prime divisors of polynomials P(T^k ); k ≥ 1

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Abstract: Let

F

be a number field,

O_{F}

the integral closure of

m a t h b b Z

in

F

and

P (T) i n O_{F} [T]

a monic separable polynomial such that

P (0) o t = 0

and

P (1) o t = 0

. We give precise sufficient conditions on a given positive integer

k

for the following condition to hold: there exist infinitely many non-zero prime ideals

m a t h c a l P

of

O_{F}

such that the reduction modulo

m a t h c a l P

of

P (T)

has a root in the residue field

O_{F} / m a t h c a l P

, but the reduction modulo

m a t h c a l P

of

P (T^{k})

has no root in

O_{F} / m a t h c a l P

. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers

k

more precise.

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A note on prime divisors of polynomials P(Tk ); k ≥ 1