# Field automorphism

In field theory, a **field automorphism** is an automorphism of the algebraic structure of a field, that is, a bijective function from the field onto itself which respects the fields operations of addition and multiplication.

The automorphisms of a given field *K* form a group, the **automorphism group** *A**u**t*(*K*).

If *L* is a subfield of *K*, an automorphism of *K* which fixes every element of *L* is termed an *L*-*automorphism*. The *L*-automorphisms of *K* form a subgroup *A**u**t*_{L}(*K*) of the full automorphism group of *K*. A field extension *K* / *L* of finite index *d* is *normal* if the automorphism group is of order equal to *d*.

## Examples

- The field
**Q**of rational numbers has only the identity automorphism, since an automorphism must map the unit element 1 to itself, and every rational number may be characterized via 1 and field operations preserved by automorphisms. - Similarly, a finite field of prime order has only the identity automorphism.
- The field
**R**of real numbers has only the identity automorphism. This is harder to prove, and relies on the fact that**R**is an ordered field, with a unique ordering defined by the positive real numbers, which are precisely the squares, so that in this case any automorphism must also respect the ordering. - The field
**C**of complex numbers has two automorphisms, the identity and complex conjugation. - A finite field
**F**_{q}of prime power order*q*, where*q*=*p*^{f}is a power of the prime number*p*, has the Frobenius automorphism, . The automorphism group in this case is cyclic of order*f*, generated by Φ. - The quadratic field has a non-trivial automorphism which maps . The automorphism group is cyclic of order 2.

A homomorphism of fields is necessarily injective, since it is a ring homomorphism with trivial kernel, and a field, viewed as a ring, has no non-trivial ideals. An endomorphism of a field need not be surjective, however. An example is the Frobenius map applied to the rational function field , which has as image the proper subfield .

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