Frobenius automorphism

In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields.

Let F be a finite field of order q, and L/F a finite extension of F defined by a polynomial irreducible over F of degree r. Then if x is an element of L,

$x \mapsto x^q$

fixes F and permutes the $q r$ elements of L in such a way as to give an automorphism of L. This is the Frobenius automorphism, also known as the Frobenius map, the Frobenius substitution, or simply the Frobenius.

In particular, if F is the field with a prime number p of elements, the Frobenius map is $x \mapsto x^p$ , fixing F and permuting L. L/F is always a Galois extension, and the corresponding Galois group is cyclic with the Frobenius automorphism as a generator.

Frobenius for local fields

The definition of the Frobenius for finite fields can be extended to other sorts of field extensions. Given an unramified finite extension L/K of local fields, there is a concept of Frobenius automorphism which induces the Frobenius automorphism in the corresponding extension of residue fields.

Suppose L/K is an unramified extension of local fields, with ring of integers O_K of K such that the residue field, the integers of K modulo their unique maximal ideal φ, is a finite field of order q. If Φ is a prime of L lying over φ, that L/K is unramified means by definition that the integers of L modulo Φ, the residue field of L, will be a finite field of order q^f extending the residue field of K where f is the degree of L/K. We may define the Frobenius map for elements of the ring of integers O_L of L by

$s_\Phi(x) \equiv x^q \ \bmod \Phi$ .

Frobenius for global fields

In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K. Since the extension is unramified the decomposition group of Φ is the Galois group of the extension of residue fields. The Frobenius then can be defined for elements of the ring of integers of L as in the local case, by

$s_\Phi(x) \equiv x^q \ \bmod \Phi$ ,

where q is the order of the residue field O_K mod φ.

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 $\Bbb{Q}_3$ gives an unramified extension $\Bbb{Q}_3(\rho)$ of $\Bbb{Q}_3$ . 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 $\Bbb{Z}_3[\rho]$ in this way; this is a polynomial of degree four in ρ with coefficients in the 3-adic integers $\Bbb{Z}_3$ . Modulo 3⁸ this polynomial is

ρ 3 + 3(460 + 183ρ - 354ρ 3 - 979ρ 3 - 575ρ 4)

This is algebraic over $\Bbb{Q}$ and is the correct global Frobenius image in terms of the embedding of $\Bbb{Q}$ into $\Bbb{Q}_3$ ; 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 $\Bbb{Q}(\beta)$ obtained by adjoining a root β satisfying

β 5 + β 4 - 4β 3 - 3β 2 + 3β + 1 = 0

to $\Bbb{Q}$ . This extension is cyclic of order five, with roots

$2 \cos \frac{2 \pi n}{11}$

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.

Frobenius automorphism

Frobenius for local fields

Frobenius for global fields

Examples

See also: Frobenius automorphism, Algebraic number theory, Automorphism, Chebyshev polynomials, Decomposition group, Discriminant, Field (mathematics), Finite extension, Finite field, Galois extension