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degree of irreducible in F_p divides the degree of the splitting field

Source: Miklos Schweitzer 2020, Problem 10

December 1, 2020

Galois Theoryabstract algebranumber theoryalgebrapolynomial

Problem Statement

Let

f

be a polynomial of degree

n

with integer coefficients and

p

a prime for which

f

, considered modulo

p

, is a degree-

k

irreducible polynomial over

\mathbb{F}_p

. Show that

k

divides the degree of the splitting field of

f

over

\mathbb{Q}