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 a 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.
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 
, the rational numbers, in the case of field characteristic zero. 
which fix a smaller field 
forms a group, by composition, called the Galois group, written 
. For example, take 
, the rational numbers, and 
. 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 
,
is a root of 
, then 
is also a root of 
.
form a field with no nontrivial automorphisms. Slightly more complicated is the extension of 
by 
, the real cube root of 2.
. 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.