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
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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 function
composition, called the Galois group, written
. For example, take 
, the rational numbers,
and
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which is an extension of 
. The two automorphisms
of 
fixing 
are the identity map and 
, where 
.
It is no accident that 
and 
are the polynomial
roots of 
. The basic observation is that for any automorphism 
of 
fixing 
, any polynomial 
with coefficients
in 
, and any field element 
,
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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.
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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.
