The Tangent Analogues of the Chebyshev Polynomials

We study the tangent analogues tan(n arctanx) of the Chebyshev polynomials from an algebraic viewpoint. They are rational functions of a pleasant form and enjoy several noteworthy properties: a useful composition law, their numerators pn(x) split into the minimal polynomials of the numbers tan kπ/n, they define the elements of the Galois groups of these minimal polynomials, and their algebraic and number theoretic properties strongly parallel those of the roots of unity. We give a complete factorization of the polynomials pn(x) in Z[x], give several applications, and list some open problems.