Perfect body
Perfect body or perfect body is a term used in algebra , in the field theory is useful because the Galois theory of perfect body avoids many complications that can occur with general bodies.
definition
A field is called perfect if all irreducible polynomials are separable , that is , if they have no multiple zeros in their decay field .
Examples
A body is perfect if and only if it is
- either has characteristic 0 (in particular the known bodies are , and perfect.)
or
- prime characteristic and the Frobenius homomorphism is an automorphism . (In particular, all finite fields are perfect.)
An example of an imperfect body is the function body for a finite body .
Equivalent characterizations
A body is perfect when it fulfills one of the following equivalent conditions.
- No over- irreducible polynomial has multiple zeros in the decay field.
- Every finite extension of is separable .
- Every algebraic extension of is separable.
- The separable conclusion of is algebraically closed .
Web links
- Perfect Field (Encyclopedia of Mathematics)
- Perfect Field (MathWorld)
Individual evidence
- ↑ Kurt Meyberg: Algebra - Part 2. Hanser 1976, ISBN 3-446-12172-2 , definition 6.9.10
- ↑ Kurt Meyberg: Algebra - Part 2. Hanser 1976, ISBN 3-446-12172-2 , sentence 6.9.11