Frobenius endomorphisms on p-adic fields¶
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class
sage.rings.padics.morphism.FrobeniusEndomorphism_padics¶ Bases:
sage.rings.morphism.RingHomomorphismA class implementing Frobenius endomorphisms on padic fields.
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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
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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
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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
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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
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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
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