Field automorphism: Difference between revisions
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* The [[quadratic field]] <math>\mathbf{Q}(\sqrt d)</math> has a non-trivial automorphism which maps <math>\sqrt d \mapsto - \sqrt d</math>. The automorphism group is cyclic of order 2. | * The [[quadratic field]] <math>\mathbf{Q}(\sqrt d)</math> has a non-trivial automorphism which maps <math>\sqrt d \mapsto - \sqrt d</math>. The automorphism group is cyclic of order 2. | ||
A homomorphism of fields is necessarily [[injective function|injective]], since it is a [[ring homomorphism]] with trivial kernel, and a field, viewed as a [[ring theory|ring]], has no non-trivial [[ideal]]s. An [[endomorphism]] of a field need not be [[surjective function|surjective]], however. An example is the Frobenius map <math>\Phi: x \mapsto x^p</math> applied to the [[rational function]] field <math>\mathbf{F}_p(X)</math>, which has as image the proper subfield <math>\mathbf{F}_p(X^p)</math>. | A homomorphism of fields is necessarily [[injective function|injective]], since it is a [[ring homomorphism]] with trivial kernel, and a field, viewed as a [[ring theory|ring]], has no non-trivial [[ideal]]s. An [[endomorphism]] of a field need not be [[surjective function|surjective]], however. An example is the Frobenius map <math>\Phi: x \mapsto x^p</math> applied to the [[rational function]] field <math>\mathbf{F}_p(X)</math>, which has as image the proper subfield <math>\mathbf{F}_p(X^p)</math>.[[Category:Suggestion Bot Tag]] |
Latest revision as of 07:01, 16 August 2024
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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 .
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 of the full automorphism group of K. A field extension 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 Fq of prime power order q, where 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 .
- 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 .