Factorization of Chebyshev Polynomials

The complete factorization of Chebyshev polynomials, of the rst and second kind, into irreducible factors over the integers Z is described. Conditions are given for determining when a Chebyshev polynomial is divisible by another. And, if non-zero, the remainder is again a Chebyshev polynomial, up to a sign. Algorithms are also speci ed to nd two in nite sets of elds Zp where a given Chebyshev polynomial factors completely into linear factors and to obtain the factors. The results also lead to the assertion: An odd integer n > 0 is prime if and only if the Chebyshev polynomial of the rst kind Tn(x), divided by x, is irreducible over the integers. Partially supported by CAPES, Brazil, under grant BEX 0744/96-4