Frobenius Endomorphism - Examples

Examples

The polynomial

x5 − x − 1

has discriminant

19 × 151,

and so is unramified at the prime 3; it is also irreducible mod 3. Hence adjoining a root ρ of it to the field of 3-adic numbers gives an unramified extension of . We may find the image of ρ under the Frobenius map by locating the root nearest to ρ3, which we may do by Newton's method. We obtain an element of the ring of integers in this way; this is a polynomial of degree four in ρ with coefficients in the 3-adic integers . Modulo 38 this polynomial is

.

This is algebraic over and is the correct global Frobenius image in terms of the embedding of into ; 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 p-adic results will suffice.

If L/K is an abelian extension of global fields, we get a much stronger congruence since it depends only on the prime φ in the base field K. For an example, consider the extension of obtained by adjoining a root β satisfying

to . This extension is cyclic of order five, with roots

for integer n. It has roots which are Chebyshev polynomials of β:

β2 - 2, β3 - 3β, β5-5β3+5β

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 22n+1 (which split). It is immediately apparent how the Frobenius map gives a result equal mod p to the p-th power of the root β.