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.