19

Part of 2021 Alibaba Global Math Competition

Problems(1)

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

Source: Alibaba Global Math Competition 2021, Problem 19

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

q

are (possible the same) prime numbers, and

a>1

is an integer, which is not a

q

algebrapolynomialsum of rootsroots of unityNested Radicalscollege contests