Field automorphism: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
m (→‎Examples: link)
imported>Richard Pinch
m (link)
Line 4: Line 4:
The automorphisms of a given field ''K'' form a [[group (mathematics)|group]], the '''automorphism group''' <math>Aut(K)</math>.
The automorphisms of a given field ''K'' form a [[group (mathematics)|group]], the '''automorphism group''' <math>Aut(K)</math>.


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 <math>Aut_L(K)</math> of the full automorphism group of ''K''.  A [[field extension]] <math>K/L</math> of finite index ''d'' is ''[[normal]]'' if the automorphism group is of [[order (group theory)|order]] equal to ''d''.   
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 <math>Aut_L(K)</math> of the full automorphism group of ''K''.  A [[field extension]] <math>K/L</math> of finite index ''d'' is ''[[normal extension|normal]]'' if the automorphism group is of [[order (group theory)|order]] equal to ''d''.   


==Examples==
==Examples==

Revision as of 12:18, 20 December 2008

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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 .

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 of the full automorphism group of K. A field extension of finite index d is normal if the automorphism group is of order equal to d.

Examples

  • 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 is a power of the prime number p, has the Frobenius automorphism, . The automorphism group in this case is cyclic of order f, generated by .
  • The quadratic field has a non-trivial automorphism which maps . 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 applied to the rational function field , which has as image the proper subfield .