Field automorphism

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In field theory, a field automorphism is an automorphism of the algebraic structure of a field, that is, a bijective function from the field onto itself which respects the fields operations of addition and multiplication.

The automorphisms of a given field K form a group, the automorphism group A u t ( K ) {\displaystyle Aut(K)} .

If L is a subfield of K, an automorphism of K which fixes every element of L is termed an L-automorphism. The L-automorphisms of K form a subgroup  A u t L ( K ) {\displaystyle Aut_{L}(K)} of the full automorphism group of K. A field extension K / L {\displaystyle K/L} of finite index d is normal if the automorphism group is of order equal to d.

## Examples[edit]

* The field Q of rational numbers has only the identity automorphism, since an automorphism must map the unit element 1 to itself, and every rational number may be obtained from 1 by field operations. which are preserved by automorphisms.
* Similarly, a finite field of prime order has only the identity automorphism.
* The field R of real numbers has only the identity automorphism. This is harder to prove, and relies on the fact that R is an ordered field, with a unique ordering defined by the positive real numbers, which are precisely the squares, so that in this case any automorphism must also respect the ordering.
* The field C of complex numbers has two automorphisms, the identity and complex conjugation.
* A finite field Fq of prime power order q, where  q = p f {\displaystyle q=p^{f}} is a power of the prime number p, has the Frobenius automorphism,  Φ : x ↦ x p {\displaystyle \Phi :x\mapsto x^{p}} . The automorphism group in this case is cyclic of order f, generated by  Φ {\displaystyle \Phi } .
* The quadratic field Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} has a non-trivial automorphism which maps  d ↦ − d {\displaystyle {\sqrt {d}}\mapsto -{\sqrt {d}}} . The automorphism group is cyclic of order 2.

A homomorphism of fields is necessarily injective, since it is a ring homomorphism with trivial kernel, and a field, viewed as a ring, has no non-trivial ideals. An endomorphism of a field need not be surjective, however. An example is the Frobenius map  Φ : x ↦ x p {\displaystyle \Phi :x\mapsto x^{p}} applied to the rational function field  F p ( X ) {\displaystyle \mathbf {F} _{p}(X)} , which has as image the proper subfield  F p ( X p ) {\displaystyle \mathbf {F} _{p}(X^{p})} .