This is true in all fields. I was playing with an idea that factors of Fermat non-primes (2^{2^n}+1 for n > 5) could be expressed in terms of products of the co

This is true in all fields. I was playing with an idea that factors of Fermat non-primes (2^{2^n}+1 for n > 5) could be expressed in terms of products of the complex factors of the polynomial x^{2^n}+1 with x=1 with the idea that it might lead to a proof that for all n > 5, F_n is composite. I spent a bunch of time working out the exact expressions for these roots and playing around with their products (which ended up having a connection to Chebyshev polynomials), and then it occurred to me to try exhaustively trying the possible products of the complex roots of x^{64}+1 with x=1 to see if it turned up the factors of F_6 and it turned out it was a dead end. A compilation of false hypotheses in different fields of mathematics could make for interesting reading (and perhaps lead to eliminating multiple people getting stuck in the same cul de sac).