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 $Aut(K)$.

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 $Aut_{L}(K)$ 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 obtained from 1 by field operations. which are 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, $\Phi :x\mapsto x^{p}$. The automorphism group in this case is cyclic of order f, generated by $\Phi$.
• The quadratic field $\mathbf {Q} ({\sqrt {d}})$ has a non-trivial automorphism which maps ${\sqrt {d}}\mapsto -{\sqrt {d}}$. 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 $\Phi :x\mapsto x^{p}$ applied to the rational function field $\mathbf {F} _{p}(X)$, which has as image the proper subfield $\mathbf {F} _{p}(X^{p})$.