moreover, the coefficients are algebraic and the result can be expressed algebraically. However, they are of degree 120, the order of the Galois group, illustrating the fact that explicit computations are much more easily accomplished if -adic results will suffice. If is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime in the base field . For an example, consider the extension of obtained by adjoining a root satisfying to . This extension is cyclic of order five, with roots for integer . It has roots which are Chebyshev polynomials of : give the result of the Frobenius map for the primes 2, 3 and 5, and so on for larger primes not equal to 11 or of the form (which split). It is immediately apparent how the Frobenius map gives a result equal mod to the -th power of the root . See also Perfect field Frobenioid Universal homeomorphism References Finite fields Algebraic number theory Galois theory