A field automorphism of a field 
is a bijective map 
that preserves all of 
's algebraic properties, more precisely, it is an isomorphism. For example, complex conjugation is a field automorphism of 
, the complex numbers, because
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
 |  |  | (4) |
A field automorphism fixes the smallest field containing 1, which is 
, the rational numbers, in the case of field characteristic zero.
The set of automorphisms of 
which fix a smaller field 
forms a group, by composition, called the Galois group, written 
. For example, take 
, the rational numbers, and
 |  |  | (5) |
 |  |  | (6) |
which is an extension of 
. Then the only automorphism of 
(fixing 
) is 
, where 
. It is no accident that 
and 
are the roots of 
. The basic observation is that for any automorphism 
, any polynomial 
with coefficients in 
, and any field element 
,
 | (7) |
So if 
is a root of 
, then 
is also a root of 
.
The rational numbers 
form a field with no nontrivial automorphisms. Slightly more complicated is the extension of 
by 
, the real cube root of 2.
 | (8) |
This extension has no nontrivial automorphisms because any automorphism would be determined by 
. But as noted above, the value of 
would have to be a root of 
. Since 
has only one such root, an automorphism must fix it, that is, 
, and so 
must be the identity map.
