Field automorphisms are important to the theory of field extensions, in particular Galois extensions.

In the case of a Galois extension L / K the subgroup of all automorphisms of L fixing K pointwise is called the Galois group of the extension.

Specifically, if L / K is a Galois extension, we consider the group G = Gal ( L / K ) consisting of all field automorphisms of L which keep all elements of K fixed.

Galois theory uses groups to describe the symmetries of the roots of a polynomial ( or more precisely the automorphisms of the algebras generated by these roots ).

The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.

the automorphisms also leave fixed, so they are elements of the Galois group.

If E is a finite Galois extension of Q < sub > p </ sub > and the building is constructed from SL < sub > n </ sub >( E ) instead of SL < sub > n </ sub >( Q < sub > p </ sub >), the Galois group Gal ( E / Q < sub > p </ sub >) will also act by automorphisms on the building.

In the above example, a connection with classical Galois theory can be seen by regarding as the profinite Galois group Gal (< span style =" text-decoration: overline "> F </ span >/ F ) of the algebraic closure < span style =" text-decoration: overline "> F </ span > of any finite field F, over F. That is, the automorphisms of < span style =" text-decoration: overline "> F </ span > fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the z < sup > n </ sup > map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n. Z of the fundamental group of the punctured disk.

Thus for K a prime field ( or ), we have but for K a field with non-trivial Galois automorphisms ( such as for or ), the projective linear group is a proper subgroup of the collineation group, which can be thought of as " transforms preserving a projective semi-linear structure ".

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E, then E / F is a Galois extension, where F is the fixed field of G.

Over an algebraically closed field, this is the only form ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 8 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 8 </ sub >)) which, because the Dynkin diagram of E < sub > 8 </ sub > ( see below ) has no automorphisms, coincides with H < sup > 1 </ sup >( k, E < sub > 8 </ sub >).

The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.

2 < sup > p </ sup >( 2 < sup > p </ sup > − 1 ) p for a prime p, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2 < sup > p </ sup >.

Its Galois group G = Gal ( K / Q ) can be determined by examining the automorphisms of K which fix a.

The outer automorphism group of a finite simple group of Lie type is an extension of a group of " diagonal automorphisms " ( cyclic except for D < sub > n </ sub >( q ) when it has order 4 ), a group of " field automorphisms " ( always cyclic ), and

For the collineation group is the projective semilinear group, – this is PGL, twisted by field automorphisms ; formally, where k is the prime field for K ; this is the fundamental theorem of projective geometry.

As with Möbius transformations, these functions can be interpreted as automorphisms of the projective line over K.

That is, the prime ideal factors of P in L form a single orbit under the automorphisms of L over K. From this and the unicity of prime factorisation, it follows that e ( j )

Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of G which contains K. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups G ( up to conjugation ).

* For any intermediate field K of E / F, the corresponding subgroup is just Aut ( E / K ), that is, the set of those automorphisms in Gal ( E / F ) which fix every element of K.

These are clearly automorphisms of K. There is also the identity automorphism e which does not change anything, and the composition of f and g which changes the signs on both radicals:

The quotient group Aut ( G ) / Inn ( G ) is usually denoted by Out ( G ); the non-trivial elements are the cosets that contain the outer automorphisms.

In other words, an automorphism of E / F is an isomorphism α from E to E such that α ( x ) = x for each x in F. The set of all automorphisms of E / F forms a group with the operation of function composition.

If E is such a subfield, write Gal ( L / E ) for the group of field automorphisms of L that hold E fixed.

is the quotient Aut ( G ) / Inn ( G ), where Aut ( G ) is the automorphism group of G and Inn ( G ) is the subgroup consisting of inner automorphisms.

* A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, ISBN 3-7643-6350-9 ( http :// emis. mi. ras. ru / journals / SC / 1997 / 2 / pdf / smf_sem-cong_2_55-81. pdf ).

If V is a vector space over the field F, the general linear group of V, written GL ( V ) or Aut ( V ), is the group of all automorphisms of V, i. e. the set of all bijective linear transformations V → V, together with functional composition as group operation.

Just as O ( n ) is the group of automorphisms which keep the quadratic polynomials x < sup > 2 </ sup > + y < sup > 2 </ sup > + ... invariant, F < sub > 4 </ sub > is the group of automorphisms of the following set of 3 polynomials in 27 variables.

F < sub > 4 </ sub > is the only exceptional lie group which gives the automorphisms of a set of real commutative polynomials.

* The study of Out ( F < sub > n </ sub >) ( the outer automorphism group of a free group of rank n ) and of individual automorphisms of free groups.

of simple groups by using the fact that F < sub > 4 </ sub > and G < sub > 2 </ sub > have extra automorphisms in characteristic 2 and 3.

If X is a Dynkin diagram, Chevalley constructed split algebraic groups corresponding to X, in particular giving groups X ( F ) with values in a field F. These groups have the following automorphisms:

Respectively, other ( non-identity ) automorphisms are called nontrivial automorphisms.

In the cases of the rational numbers ( Q ) and the real numbers ( R ) there are no nontrivial field automorphisms.

In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely ( uncountably ) many " wild " automorphisms ( assuming the axiom of choice ).

For example, the automorphisms of the Riemann sphere are Möbius transformations.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself.

The other automorphisms are called outer automorphisms.

For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to < var > S </ var >< sub > 3 </ sub > and Dih < sub > 3 </ sub >.

For Abelian groups all automorphisms except the trivial one are called outer automorphisms.

Isomorphisms, automorphisms, and endomorphisms are all types of homomorphism.

For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same ( here " the same " does not mean something like e. g. " the same up to translation and rotation ", but it means " exactly the same ").

* The space is a cube with Euclidean metric ; the figures include cubes of the same size as the space, with colors or patterns on the faces ; the automorphisms of the space are the 48 isometries ; the figure is a cube of which one face has a different color ; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure.

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group.

The outer automorphism group measures, in a sense, how many automorphisms of G are not inner.

At the opposite end of the spectrum, it is possible that the inner automorphisms exhaust the entire automorphism group ; a group whose automorphisms are all inner and whose centre is trivial is called complete.

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.