Frobenius endomorphisms on \(p\)-adic fields¶
- class sage.rings.padics.morphism.FrobeniusEndomorphism_padics[source]¶
- Bases: - RingHomomorphism- A class implementing Frobenius endomorphisms on \(p\)-adic fields. - is_identity()[source]¶
- Return - Trueif 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 - >>> from sage.all import * >>> K = Qq(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> Frob = K.frobenius_endomorphism() >>> Frob.is_identity() False >>> (Frob**Integer(3)).is_identity() True 
 - is_injective()[source]¶
- Return - Truesince any power of the Frobenius endomorphism over an unramified \(p\)-adic field is always injective.- EXAMPLES: - sage: K.<a> = Qq(5^3) sage: Frob = K.frobenius_endomorphism() sage: Frob.is_injective() True - >>> from sage.all import * >>> K = Qq(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> Frob = K.frobenius_endomorphism() >>> Frob.is_injective() True 
 - is_surjective()[source]¶
- Return - Truesince any power of the Frobenius endomorphism over an unramified \(p\)-adic field is always surjective.- EXAMPLES: - sage: K.<a> = Qq(5^3) sage: Frob = K.frobenius_endomorphism() sage: Frob.is_surjective() True - >>> from sage.all import * >>> K = Qq(Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1) >>> Frob = K.frobenius_endomorphism() >>> Frob.is_surjective() True 
 - order()[source]¶
- 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 - >>> from sage.all import * >>> K = Qq(Integer(5)**Integer(12), names=('a',)); (a,) = K._first_ngens(1) >>> Frob = K.frobenius_endomorphism() >>> Frob.order() 12 >>> (Frob**Integer(2)).order() 6 >>> (Frob**Integer(9)).order() 4 
 - power()[source]¶
- 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 - >>> from sage.all import * >>> K = Qq(Integer(5)**Integer(12), names=('a',)); (a,) = K._first_ngens(1) >>> Frob = K.frobenius_endomorphism() >>> Frob.power() 1 >>> (Frob**Integer(9)).power() 9 >>> (Frob**Integer(13)).power() 1