Computation of the Cubic Root of a p-adic Number

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Introduction
Let Q p be the field of p-adic numbers where p is a prime number.Our main goal is to compute the approximate finite p-adic expansion of the cubic root for the p-adic number a ∈ Q * p .This is done by determining the approximate solution of the equation (1) The solution of (1) is approximated by a p-adic number sequence (x n ) n ⊂ Q * p constructed by the secant method.Knapp and Xenophotos (2010) used numerical methods to find the reciprocal of an integer modulo p n .

Preliminaries
Definition 3. Let p be a prime number.The field Q p of p-adic numbers is the completion of the field Q of rational numbers with respect to the p-adic norm |•| p defined by where v p is the p-adic valuation defined by v p (x) = max {r ∈ Z : p r | x}.
The p-adic norm induces a metric d p given by this metric is called the p-adic metric.
The short representation of a is β n β n+1 ...β −1 • β 0 β 1 ..., where only the coefficients of the powers of p are shown.We can use the p-adic point • as a device for displaying the sign of n as follows: Definition 5. A p-adic number a ∈ Q p is said to be a p-adic integer if this canonical expansion contains only non negative power of p.
The set of p-adic integers is denoted by Z p .We have Definition 6.A p-adic integer a ∈ Z p is said to be a p-adic unit if the first digit β 0 in the p-adic expansion is different of zero.The set of p-adic units is denoted by Z * p .Hence we have Vej, 2000) Given a p-adic number a ∈ Q p \ {0}, there exist n ∈ Z and u ∈ Z * such that a = p n • u.
Proposition 9. (S.Katok, 2007)  We use the notation H(p, M, a) where p is a prime and M is the integer which specifies the number of precision digits of the p-adic expansion.
Theorem 11. (Hensel's lemma) (S.Katok, 2007) Let F(x) = c 0 + c 1 x + ... + c n x n be a polynomial whose coefficients are p-adic integers i.e.F ∈ Z be the derivative of F(x).Suppose a 0 is a p-adic integer which satisfies F(a 0 ) ≡ 0( mod p) and F (a 0 ) 0( mod p).Then there exists a unique p-adic integer a such that F(a) = 0 and a ≡ a 0 ( mod p).
Theorem 12. (S.Katok, 2007) A polynomial with integer coefficients has a root in Z p if and only if it has an integer root modulo p k for any k 1.

Main Results
Proposition 14.A rational integer a not divisible by p has a cubical root in Z p (p 3) if and only if a is a cubic residue modulo p.
Corollary 15.Let p be a prime number, then Let us note that u and v are p-adic unit intergers according to definition 4 Then, imposing the cubic condition, we obtain The latter is equivalent to the following system Additionally, we consider f (x) = x 3 0 − a and its derivative f Then we have 1. if p 3, by Hensel's lemma the solution of f and and a 1 = 2.
Let a ∈ Q * p be a p-adic number such that We know that if there exists a p-adic number d such that d 3 = a and (x n ) n is a sequence of the p-adic numbers that converges to a p-adic number d 0, then from a certain rank one has

The secant method
An elementary method to determine zeros of a given function is the secant method.This method can be derived from the Newton method, where we replace the derivative f (x n ) by the approximation The iterative formula of the secant method is Obtaining the following recurrence relation Determining the rate of convergence of an iterative method is to study the comportment of the sequence (e n+n 0 ) n defined by e n+n 0 = x n+n 0 − x n+n 0 −1 obtained at each step of the iteration where n 0 ∈ N.
Roughly speaking, if the rate of convergence of a method is s, then after each iteration the number of correct significant digits in the approximation increases by a factor of approximately s.
Theorem 16.If x n 0 −1 is the cubic root of a of order α and x n 0 is the cubic root of a of order β then 1.If p 3, then x n+n 0 −1 is the cubic root of a of order J n , where the sequence (J n ) n is defined by 2. If p = 3, then x n+n 0 −1 is the cubic root of a of order J n , where the sequence (J n ) n is defined by were is the golden ratio and m is the exponent in the p-adic expansion of a.
Published by Canadian Center of Science and Education Proof.Let (x n ) n be the sequence defined by (13).Then This gives and, hence, we have This is equivalent to verify either Or, in virtue of lemma 5 On the other hand, we have In this manner, we find that if p 3, then x 3 n 0 +4 − a ≡ 0( mod 3 3α+5β−21(m+1) ) . . .
where the sequence (J n ) n is defined by Where and The sequences (F n ) n and (A n ) n are linear recurrent sequences whose general terms are given respectively by and We obtain 2. If p = 3, then where the sequence (J n ) n is defined by Corollary 17. Suppose that x n 0 −1 is the cubic root of a of order α and that x n 0 is the cubic root of a of order β then 1.If p 3, then x n+n 0 − x n+n 0 −1 ≡ 0( mod p λ n ) where the sequence (λ n ) n is defined by 2. If p = 3, then x n+n 0 − x n+n 0 −1 ≡ 0( mod 3 λ n ) where the sequence λ n n is defined by Proof.Starting from equation ( 13), we have We obtain x n+n 0 − x n+n 0 −1 ≡ 0( mod p λ n ), if p 3, x n+n 0 − x n+n 0 −1 ≡ 0( mod 3 λ n ), si p = 3.
Such as

Conclusions
According to the results obtained in the previous section, we obtain the following conclusions: 1.If p 3,then (a) The rate of convergence of the sequence (x n ) n is of order λ n .
Let (a n ) n be a p-adic number sequence.If lim