Factorization properties of chebyshev polynomials

Several factorization properties of Chebyshev are reported here. Studying the euclidean division of two Chebyshev, we observe that the remainder is either zero or (up to a sign) another Chebyshev polynomial. This lead to explicit computation of the greatest common divisor of two Chebyshev. We also obtain conditions for determining when a Chebyshev polynomial is divisible by another. Observing the modular representation over prime fields, we find two infinite sets of fields p where a given Chebyshev polynomial factors completely into linear factors. We discuss how to obtain the factors.