Body homomorphism
In mathematics , especially in algebra , a body homomorphism is a structure-preserving mapping between bodies .
definition
Be and two bodies.
- such as
It is therefore irrelevant whether elements are first linked in and the result is then mapped by a homomorphism, or whether the linking of the corresponding function values is only done in.
- A bijective body homomorphism is called a body isomorphism .
Bodies, between which an isomorphism exists, in signs , are indistinguishable from the point of view of (abstract) algebra .
- A body isomorphism of a body in itself is called body automorphism .
The Galois theory deals specifically with body automorphisms that leave a given sub-body invariant.
properties
- In particular, each body is a ring with a one . Correspondingly, a body homomorphism is only a ring homomorphism for which it is additionally required that the following applies. In particular, it induces a group homomorphism of the additive groups as well as a group homomorphism of the multiplicative groups.
- A body homomorphism is always injective : Since the core of a ring homomorphism is an ideal , but the body only has the trivial ideals and , must therefore apply. Hence is injective.
- A Körperautomorphismus always be at least the prime field of invariant.
Examples
- The complex conjugation is a field automorphism of the field of complex numbers , which leaves the subfield of real numbers invariant.
- For a body whose characteristic is a prime number , the Frobenius homomorphism is a body endomorphism that leaves a sub-body that is too isomorphic. If the body is finite, this mapping is even a body automorphism.
- Prime bodies , for example , have no body automorphisms with the exception of identity mapping.
literature
- Siegfried Bosch : Algebra. 6th edition. Springer, Berlin / Heidelberg 2006, ISBN 3-540-40388-4 .
- Falko Lorenz : Introduction to Algebra. Part I. Bibliographisches Institut, Mannheim 1987, ISBN 3-411-03171-9 .