ABSTRACT

Recall that in [1] it is obtained the criteria solvability of the Equation in and for. Since any p-adic number x has a unique form where and in [1] it is also shown that from the criteria in it follows the criteria in and. In this paper we provide the algorithm of finding the solutions of the Equation in with coefficients from.

Keywords:p-Adic Numbers; Solvability of Equation; Congruence

1. Introduction

In the present time description of different structures in mathematics are studying over field of -adic numbers. In particular, -adic analysis is one of the intensive developing directions of modern mathematics. Numerous applications of -adic numbers have found their own reflection in the theory of -adic differential equations, -adic theory of probabilities, -adic mathematical physics, algebras over - adic numbers and others.

The field of -adic numbers were introduced by German mathematician K. Hensel at the end of the 19th century [2]. The investigation of -adic numbers were motivated primarily by an attempt to bring the ideas and techniques of the power series into number theory. Their canonical representation is similar to expansion of analytical functions in power series, which is analogy between algebraic numbers and algebraic functions. There are several books devoted to study -adic numbers and -adic analysis [3-6].

Classification of algebras in small dimensions plays important role for the studying of properties of varieties of algebras. It is known that the problem of classification of finite dimensional algebras involves a study on equations for structural constants, i.e. to the decision of some systems of the Equations in the corresponding field. Classifications of complex Leibniz algebras have been investigated in [7-10] and many other works. In similar complex case, the problem of classification in -adic case is reduced to the solution of the Equations in the field. The classifications of Leibniz algebras over the field of -adic numbers have been obtained in [11-13].

In the field of complex numbers the fundamental Abel’s theorem about insolvability in radicals of general Equation of -th degree is well known. In this field square equation is solved by discriminant, for cubic Equation Cardano’s formulas were widely applied. In the field of -adic numbers square equation does not always has a solution. Note that the criteria of solvability of the Equation is given in [6,14,15] we can find the solvability criteria for the Equation where is an arbitrary natural number.

In this paper we consider adic cubic equation By replacing this equation become the so-called depressed cubic equation

(1)

The solvability criterion for the cubic equation over -adic numbers is different from the case Note that solvability criteria for is obtained in [1]. The problem of finding a solvability criteria of the cubic equation for the case is complicated. This problem was partially solved in [16], namely, it is obtained solvability criteria of cubic equation with condition.

In this paper we obtain solvability criteria of cubic equation for without any conditions. Moreover, the algorithm of finding the solutions of the equation in with coefficients from is provided.

2. Preliminaries

Let be a field of rational numbers. Every rational number can be represented by the form

, where is a positive integer, and is a fixed prime number. In a norm has been defined as follows:

The norm is called a -adic norm of and it satisfies so called the strong triangle inequality. The completion of with respect to -adic norm defines the -adic field which is denoted by ([4,6]). It is well known that any -adic number can be uniquely represented in the canonical form

where and are integers, , ,. -Adic number for which, is called integer -adic number, and the set of such numbers is denoted by Integer

for which is called unit of and their set is denoted by.

For any numbers and it is known the following result.

Theorem 2.1 [3]. If then a congruence has one and only one solution.

We also need the following Lemma.

Lemma 2.1 [14]. The following is true:

where, , and for

From Lemma 2.1 by we have

For we put

Also the following identity is true:

(2)

3. The Main Result

In this paper we study the cubic Equation (1) over the field -adic numbers, i.e.

Put

where.

Since any -adic number has a unique form where and we will be limited to search a decision from i.e..

Putting the canonical form of and in (1), we get

By Lemma 2.1 and Equality (2), the Equation (1) becomes to the following form:

(3)

Proposition 3.1 If one of the following conditions:

is fulfilled, then the Equation (1) has not a solution in

Proof. 1) Let and Multiplying Equation (3) by we get the following congruence which is not correct. Consequently, Equation (1) has no solution in

2) Let and Then from (3) it follows a congruence which has no a nonzero solution.Therefore, in Equation (1) does not have a solution.

In other cases, we analogously get the congruences

which are not hold. Therefore, in there is no solution.■

From the Proposition 3.1 we have that the cubic equation may have a solution if one of the following four cases

is hold.

In the following theorem we present an algorithm of finding of the Equation for the first case.

Theorem 3.1 Let and Then to be a solution of the Equation (1) in if and only if the congruences

are fulfilled, where integers are defined consequently from the following correlations

Proof. Let

is a solution of Equation (1), then Equality (3) becomes

So we have

from which it follows the necessity in fulfilling the congruences of the theorem.

Now let is satisfied the congruences of the theorem. Since then by Theorem 2.1 it implies that these congruences have the solutions

Then

Therefore, we show that is a solution of the Equation (1).■

Let us examine a case and get necessary and sufficient conditions for a solution of Equation (1).

Theorem 3.2 Let and Then to be a solution of Equation (1) in if and only if the congruences

are fulfilled, where integers are defined consequently from the following correlations

Proof. Let is a solution of the Equation (1), then Equality (3) becomes

Therefore, we have

from which it follows the necessity in fulfilling the congruences of the theorem.

Now let is satisfied the congruences of the theorem. Since then by Theorem 2.1 there are solutions of the congruences.

Putting element to Equality (3), we have

Therefore, we show that is a solution of Equation (1).■

The following theorem gives necessary and sufficient conditions for a solution of Equation (1) for the case and

Theorem 3.3 Let

Then to be a solution of Equation (1) in if and only if the next congruences

are fulfilled, where integers are defined consequently from the equalities

Proof. The proof of the Theorem can be obtained by similar way to the proofs of Theorems 3.1 and 3.2.■

Examining various cases of and we need to study only the case and Because of appearance of uncertainty of a solution, we divide this case to and

Theorem 3.4 Let and or Then to be a solution of Equation (1) in if and only if he next congruences

are fulfilled, where integers are defined from the equalities

Proof. Analogously to the proof of Theorem 3.1.■

Theorem 3.5 Let and or Then to be a solution of the Equation (1) in if and only if he next congruences

are fulfilled, where integers are defined from the equalities

Proof. Analogously to the proof of Theorem 3.1.■

Similarly to Theorem 3.4, it is proved the following Theorem 3.6 Let and or Then to be a solution of the Equation (1) in if and only if he next congruences

are fulfilled, where integers are defined from the equalities

Now we consider Equality (3) with Put

Theorem 3.7 Let and to be so that Then to be a solution of Equation (1) in if and only if the congruences

are faithfully, where and integers are defined from the equalities

Proof. Let the congruences has a solution Then denote by

the number satisfying the equality

Using Theorem 2.1, we have existence of solutions of the congruences

The next chain of equalities

shows that is a solution of Equation (1).■

From the proof of Theorem 3.7, it is easy to see that if then we have the following congruences and appropriate equalities a) i.e.

b) then

c) then

(4)

d) it follows that

Since then the congruence d) can be written in the form

and so we have

If for any natural number we have then we could establish the criteria of solvability for Equation (1). However, if there exists, such that then the criteria of solvability can be found, and therefore, we need the following Lemma 3.1 Let and to be so that for some fixed. If be a solution of Equation (1), then it is true the following system of the congruences

(5)

where and integers are defined from the equalities

(6)

Proof. We will prove Theorem by induction. Let i.e.

then the system of the congruences (4) are true. Note that

From (3) it is easy to get

Therefore,

(7)

where

Obviously, the statement of Lemma is true for i.e. for

Let i.e., then from the equalities (7) it follows that the following congruences are be added to the system (4):

e) it follows

f) it follows

where and are defined by equalities

Since we denote by and have e)

f)

h) where

So we showed that the statement of Lemma is true for

Let the system of congruences (5) and (6) is true for Since then from the congruences

we derive

It is easy to check that

By these correlations we deduce

For we get

Consequently, we have

So we established that the system of congruences (5)-(6) is true for■

Using the Lemma 3.1 we obtain the following Theorems.

Theorem 3.8 Let and to be such that for some fixed. Then to be a solution of the Equation (1) in if and only if the system of the congruences

has a solution, where and integers are defined from the equalities

Theorem 3.9 Let and to be so that for all Then to be a solution of the Equation (1) in if and only if the system of the congruences

has a solution, where and integers are defined from the equalities

Acknowledgements

The first author was supported by grant UniKL/IRPS/str11061, Universiti Kuala Lumpur.

REFERENCES