# Frobenius endomorphisms on p-adic fields#

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