Derivations of function fields

For global function fields, which have positive characteristics, the higher derivation is available:

sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]
sage: L.<y> = K.extension(Y^3 + x + x^3*Y)                                          # needs sage.rings.function_field
sage: h = L.higher_derivation()                                                     # needs sage.rings.function_field
sage: h(y^2, 2)                                                                     # needs sage.rings.function_field
((x^7 + 1)/x^2)*y^2 + x^3*y

AUTHORS:

• William Stein (2010): initial version

• Julian Rüth (2011-09-14, 2014-06-23, 2017-08-21): refactored class hierarchy; added derivation classes; morphisms to/from fraction fields

• Kwankyu Lee (2017-04-30): added higher derivations and completions

class sage.rings.function_field.derivations.FunctionFieldDerivation(parent)

Bases: RingDerivationWithoutTwist

Base class for derivations on function fields.

A derivation on $R$ is a map $R\to R$ with $D(\alpha +\beta )=D(\alpha )+D(\beta )$ and $D(\alpha \beta )=\beta D(\alpha )+\alpha D(\beta )$ for all $\alpha ,\beta \in R$.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: d
d/dx

is_injective()

Return False since a derivation is never injective.

EXAMPLES:

sage: K.<x> = FunctionField(QQ)
sage: d = K.derivation()
sage: d.is_injective()
False