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.