Rigid body (algebra)

A rigid field (English: rigid field ) is an excellent algebraic structure in the mathematical sub-area of algebra , namely a body that only allows a single one, the trivial one, namely identity, as a (body) automorphism .

Examples

A prime body is rigid. Because for every automorphism is in included and a body (the fixed field ). Since it does not contain a real partial body, the fixed body is equally whole and has a trivial effect . ${\displaystyle \Pi }$ ${\displaystyle \sigma }$ ${\displaystyle \operatorname {Fix} (\Pi ,\sigma ):=\{x\in \Pi \;|\;\sigma (x)=x\}}$ ${\displaystyle \Pi }$ ${\displaystyle \Pi }$ ${\displaystyle \Pi }$ ${\displaystyle \sigma }$ ${\displaystyle \Pi }$

The rigid bodies of characteristic 0 are exactly the Euclidean bodies . These include a. the prime field of rational numbers , the field of real numbers and the real closed field of algebraic real numbers. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {A} \cap \mathbb {R} }$

Counterexamples

An intermediate body is not automatically rigid if the upper and part of the body are. E.g. the square number field , which lies between the rational numbers and the real numbers ( ), has a non-trivial conjugation mapping . ${\displaystyle \mathbb {Q} ({\sqrt {2}})}$ ${\displaystyle \mathbb {Q} \subset \mathbb {Q} ({\sqrt {2}})\subset \mathbb {A} \!\cap \!\mathbb {R} }$

A field of characteristic 0, an element with contains also contains a Konjugationsabbildung is therefore not rigid. ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$

Individual evidence

1. ^ Albrecht Beutelspacher : Lineare Algebra . 7th edition. Vieweg + Teubner Verlag , Wiesbaden 2010, ISBN 978-3-528-66508-1 , p. 40-41 .