Frobenius endomorphisms on p-adic fields¶
-
class
sage.rings.padics.morphism.
FrobeniusEndomorphism_padics
¶ Bases:
sage.rings.morphism.RingHomomorphism
A class implementing Frobenius endomorphisms on padic fields.
-
is_identity
()¶ Return true if this morphism is the identity morphism.
EXAMPLES:
sage: K.<a> = Qq(5^3) sage: Frob = K.frobenius_endomorphism() sage: Frob.is_identity() False sage: (Frob^3).is_identity() True
-
is_injective
()¶ Return true since any power of the Frobenius endomorphism over an unramified padic field is always injective.
EXAMPLES:
sage: K.<a> = Qq(5^3) sage: Frob = K.frobenius_endomorphism() sage: Frob.is_injective() True
-
is_surjective
()¶ Return true since any power of the Frobenius endomorphism over an unramified padic field is always surjective.
EXAMPLES:
sage: K.<a> = Qq(5^3) sage: Frob = K.frobenius_endomorphism() sage: Frob.is_surjective() True
-
order
()¶ Return the order of this endomorphism.
EXAMPLES:
sage: K.<a> = Qq(5^12) sage: Frob = K.frobenius_endomorphism() sage: Frob.order() 12 sage: (Frob^2).order() 6 sage: (Frob^9).order() 4
-
power
()¶ Return the smallest integer \(n\) such that this endomorphism is the \(n\)-th power of the absolute (arithmetic) Frobenius.
EXAMPLES:
sage: K.<a> = Qq(5^12) sage: Frob = K.frobenius_endomorphism() sage: Frob.power() 1 sage: (Frob^9).power() 9 sage: (Frob^13).power() 1
-