Body homomorphism

In mathematics , especially in algebra , a body homomorphism is a structure-preserving mapping between bodies .

definition

Be and two bodies. ${\displaystyle (K;+_{K};*_{K})}$ ${\displaystyle (L;+_{L};*_{L})}$

• A function is called body homomorphism if it satisfies the following axioms : ${\displaystyle f\colon K\to L}$

1. ${\displaystyle f(0_{K})=0_{L}}$ such as ${\displaystyle f(1_{K})=1_{L}}$
2. ${\displaystyle \forall a;b\in K\colon f(a+_{K}b)=f(a)+_{L}f(b)}$
3. ${\displaystyle \forall a;b\in K\colon f(a*_{K}b)=f(a)*_{L}f(b)}$

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. ${\displaystyle K}$ ${\displaystyle L}$

• 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 . ${\displaystyle K\cong L}$

• A body isomorphism of a body in itself is called body automorphism . ${\displaystyle f\colon K\to K}$

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. ${\displaystyle f\colon K\to L}$ ${\displaystyle f(1)=1}$ ${\displaystyle f}$ ${\displaystyle f\colon (K,+)\to (L,+)}$ ${\displaystyle f\colon (K\setminus \{0\},\cdot )\to (L\setminus \{0\},\cdot )}$
• 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. ${\displaystyle f\colon K\to L}$ ${\displaystyle K}$ ${\displaystyle \{0\}}$ ${\displaystyle K}$ ${\displaystyle f(1)\neq 0}$ ${\displaystyle \ker f=\{0\}}$ ${\displaystyle f}$
• A Körperautomorphismus always be at least the prime field of invariant. ${\displaystyle f\colon K\to K}$ ${\displaystyle K}$

Examples

• The complex conjugation is a field automorphism of the field of complex numbers ${\displaystyle \mathbb {C} }$ , which leaves the subfield of real numbers ${\displaystyle \mathbb {R} }$ 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. ${\displaystyle p}$ ${\displaystyle x\mapsto x^{p}}$ ${\displaystyle \mathbb {F} _{p}}$
• Prime bodies , for example , have no body automorphisms with the exception of identity mapping. ${\displaystyle \mathbb {F} _{p}}$

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 .