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 composition,
called the Galois group, written 
. For example, take 
, the rational numbers, and
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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 
,
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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.
