MathDB

For which p,q is this nested radical a sum of roots of unity?

Source: Alibaba Global Math Competition 2021, Problem 19

July 4, 2021

algebrapolynomialsum of rootsroots of unityNested Radicalscollege contests

Problem Statement

Find all real numbers of the form

\sqrt[p]{2021+\sqrt[q]{a}}

that can be expressed as a linear combination of roots of unity with rational coefficients, where

p

and

q

are (possible the same) prime numbers, and

a>1

is an integer, which is not a

q

-th power.