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 . ${\displaystyle K}$

Examples

A body is perfect if and only if it is

• either has characteristic 0 (in particular the known bodies are , and perfect.) ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

or

• prime characteristic and the Frobenius homomorphism is an automorphism . (In particular, all finite fields are perfect.) ${\displaystyle p}$

An example of an imperfect body is the function body for a finite body . ${\displaystyle \mathbb {F} _{q}(X)}$ ${\displaystyle \mathbb {F} _{q}}$

Equivalent characterizations

A body is perfect when it fulfills one of the following equivalent conditions. ${\displaystyle K}$

• No over- irreducible polynomial has multiple zeros in the decay field. ${\displaystyle K}$
• Every finite extension of is separable . ${\displaystyle K}$
• Every algebraic extension of is separable. ${\displaystyle K}$
• The separable conclusion of is algebraically closed . ${\displaystyle K}$

Web links

• Perfect Field (Encyclopedia of Mathematics)
• Perfect Field (MathWorld)

Individual evidence

1. ↑ Kurt Meyberg: Algebra - Part 2. Hanser 1976, ISBN 3-446-12172-2 , definition 6.9.10
2. ↑ Kurt Meyberg: Algebra - Part 2. Hanser 1976, ISBN 3-446-12172-2 , sentence 6.9.11