Question

A nonzero polynomial with rational coefficients has all of the numbers [1 sqrt{2}, ; 2 sqrt{3}, ;3 sqrt{4},; dots, ;1000 sqrt{1001}]as roots. What is the smallest possible degree of such a polynomial?

Answer 1

Its degree can be at least 1970

Explanation:

for each root of the form √q, where q is not a square, we have a root -√q. Therefore, we need to find, among the numbers below to 1000, how many sqaures there are.

Since √1000 = 31.6, we have a total of 30 squares:

2², 3², 4², ...., 30², 31²

Each square gives one root and the non squares (there are 1000-30 = 970 of them) gives 2 roots (one for them and one for the opposite). Hence the smallest degree a rational polynomial can have is

970*2 + 30 = 1970