+159 A nonzero polynomial with rational coefficients has all of the numbers 1+√2,2+√3,3+√4,…,1000+√1001as roots. What is the smallest possible degree of such a polynomial?
+6252 The polynomial evaluated at 1 is the sum of the coefficients. This must be a rational numberThus each root that is not a rational number must have it's conjugate also as a rootThis will be all the roots of the form a+√b where √b∉{2,3,…,1001}
There are 30 numbers in 2-1001 that are perfect squaresSo 1000−30=970 of the roots must also have their conjugate as a rootThus we have a total of 30+2⋅970=1970 is the smallest possible degree of the polynomial
I'm pretty sure this is correct but I might be missing something. Someone else should take a look.
