Alied Mathematics 14 5 5-46 Published Online January 14 (htt://wwwscirorg/ournal/am) htt://ddoiorg/146/am14515 On the Solutions of the Equation + A = B in with Coefficients from I M Rihsiboev 1 A Kh Khudoyberdiyev T K Kurbanbaev K K Masutova 1 Universiti Kuala Lumur Malaysian Institute of Industrial Technology Johor Bahru Malaysia Institute of Mathematics Tashent Uzbeistan Email: iromr@gmailcom habror@mailru tuuelbay@mailru amilyam81@mailru Received August 1 1; revised Setember 1; acceted October 7 1 Coyright 14 I M Rihsiboev et al This is an oen access article distributed under the Creative Commons Attribution License which ermits unrestricted use distribution and reroduction in any medium rovided the original wor is roerly cited In accordance of the Creative Commons Attribution License all Coyrights 14 are reserved for SCIRP and the owner of the intellectual roerty I M Rihsiboev et al All Coyright 14 are guarded by law and by SCIRP as a guardian ABSTRACT Recall that in [1] it is obtained the criteria solvability of the Equation > Since any -adic number has a unique form shown that from the criteria in algorithm of finding the solutions of the Equation KEYWORDS it follows the criteria in -Adic Numbers; Solvability of Equation; Congruence 1 Introduction + a = b in = where and + a = b in and for and in [1] it is also In this aer we rovide the with coefficients from In the resent time descrition of different structures in mathematics are studying over field of -adic numbers In articular -adic analysis is one of the intensive develoing directions of modern mathematics Numerous alications of -adic numbers have found their own reflection in the theory of -adic differential equations -adic theory of robabilities -adic mathematical hysics 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 [] The investigation of -adic numbers were motivated rimarily by an attemt to bring the ideas and techniques of the ower series into number theory Their canonical reresentation is similar to eansion of analytical functions in ower series which is analogy between algebraic numbers and algebraic functions There are several boos devoted to study -adic numbers and -adic analysis [-6] Classification of algebras in small dimensions lays imortant role for the studying of roerties of varieties of algebras It is nown that the roblem of classification of finite dimensional algebras involves a study on equations for structural constants ie to the decision of some systems of the Equations in the corresonding field Classifications of comle Leibniz algebras have been investigated in [7-1] and many other wors In similar comle case the roblem 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 [11-1] In the field of comle numbers the fundamental Abel s theorem about insolvability in radicals of general Equation of n -th degree ( n >5) is well nown In this field square equation is solved by discriminant for cubic Equation Cardano s formulas were widely alied In the field of -adic numbers square equation does
6 I M RIKHSIBOEV ET AL not always has a solution Note that the criteria of solvability of the Equation a is given in [61415] we q can find the solvability criteria for the Equation a where q is an arbitrary natural number r In this aer we consider adic cubic equation y ry syt By relacing y this equa- tion become the so-called deressed cubic equation a b (1) The solvability criterion for the cubic equation a b over -adic numbers is different from the case > Note that solvability criteria for > is obtained in [1] The roblem of finding a solvability criteria of the cubic equation for the case is comlicated This roblem was artially solved in [16] namely it is 1 obtained solvability criteria of cubic equation with condition a In this aer we obtain solvability criteria of cubic equation for without any conditions Moreover the algorithm of finding the solutions of the equation a b in with coefficients from is rovided Preliminaries Let be a field of rational numbers Every rational number can be reresented by the form n where n m is a ositive integer n 1 m 1 and is a fied rime m 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 comletion of with resect to -adic norm defines the -adic field which is denoted by ([46]) It is well nown that any -adic number can be uniquely reresented in the canonical form 1 where and are integers 1 1 -Adic number for which 1 is called integer -adic number and the set of such numbers is denoted by Integer for which 1 is called unit of and their set is denoted by For any numbers a and m it is nown the following result Theorem 1 [] If am 1 then a congruence a bmodm has one and only one solution We also need the following Lemma Lemma 1 [14] The following is true: q i q q i q N 1 1 i 1 1 where 1 N1 and for From Lemma 1 by q we have For we ut q! N N m m1 m1 1 1 1 m m1 m 1: m! m1! m 1! 1 1 imiq iimi i i N 1 1 i 1
I M RIKHSIBOEV ET AL 7 6 P P m m m 1 1 1 1 6 1 1 m m1 m 1: m! m1! m 1! 1 1 imi i1imi Also the following identity is true: i a i a s s i s () The Main Result In this aer we study the cubic Equation (1) over the field -adic numbers ie ab Put a b 1 1 1 a a a b b b where a b 1 a b 1 Since any -adic number has a unique form where and we will be limited to search a decision from ie Putting the canonical form of ab and in (1) we get a b a b By Lemma 1 and Equality () the Equation (1) becomes to the following form: a N( 1 1) a sas 1 1 s b b b 1 Proosition 1 If one of the following conditions: a b a b a b a b a b 1) and < ; ) > and > ; ) > and < ; 4) < and ; 5) < and > () is fulfilled then the Equation (1) has not a solution in b Proof 1) Let a and b < Multilying Equation () by we get the following con- gruence b mod which is not correct Consequently Equation (1) has no solution in ) Let a > and b > Then from () it follows a congruence mod which has no a nonzero solutiontherefore in Equation (1) does not have a solution In other cases we analogously get the congruences a b mod or mod which are not hold Therefore in there is no solution From the Proosition 1 we have that the cubic equation may have a solution if one of the following four cases a b a b a b a b 1) ) ) 4) is hold In the following theorem we resent an algorithm of finding of the Equation a b for the first case a b and a 1 Then to be a solution of the Equation (1) in if and Theorem 1 Let
8 I M RIKHSIBOEV ET AL only if the congruences a b mod a a N M b 1 1 1 1 1 mod a a N M b mod 1 1 1 1 1 are fulfilled where integers 1 are defined consequently from the following correlations M 1 a b M a a N b M M 1 1 1 1 1 1 a a a N 1 1 1 1 1 b M M 1 1 1 1 1 Proof Let 1 1 is a solution of Equation (1) then Equality () becomes So we have N 1 1 a sas b b 1 1 s 1 1 1 1 1 1 1 1 1 1 a a a N a a a N b b from which it follows the necessity in fulfilling the congruences of the theorem Now let is satisfied the congruences of the theorem Since a 1 then by Theorem 1 it imlies that these congruences have the solutions Then N 1 1 a sas 1 1 s 1 1 1 1 1 1 1 1 1 1 a N a a N a a a a 1 b M b M M b M M b 1 1 1 1 1 1 1 b 1 Therefore we show that Let us eamine a case a b Equation (1) Theorem Let a b m > and only if the congruences is a solution of the Equation (1) > and get necessary and sufficient conditions for a solution of and a Then to be a solution of Equation (1) in if
I M RIKHSIBOEV ET AL 9 a mod 1 1 1 1 mod a a N M a a N M mod m1 1 1 1 a a N M b mod m 1 1 1 m are fulfilled where integers M 1 1 1 1 1 1 M 1 a a N M M 1 1 1 1 1 a a a N 1 1 1 1 1 M M m1 a a a N 1 1 1 1 1 b M M m 1 1 1 1 are defined consequently from the following correlations m Proof Let is a solution of the Equation (1) then Equality () becomes m N 1 1 a sas b b 1 1 s 1 Therefore we have 1 1 1 1 1 1 1 a a a N a a a N m b b 1 from which it follows the necessity in fulfilling the congruences of the theorem Now let is satisfied the congruences of the theorem Since a 1 then by Theorem 1 there are solutions of the congruences Putting element to Equality () we have N 1 1 a sas 1 1 s 1 1 1 1 1 1 1 1 1 1 a N aa N a a aa m1 M M M M M 1 1 1 1 1 1 1 b M M m m 1 1 1 1 m b b 1 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 a < b Theorem Let and < a b m m Then to be a solution of Equation (1) in if and only if the net congruences
4 I M RIKHSIBOEV ET AL a bmod a 1a1a M 1 1b mod 1 m1 ma m 1a1 am Mm 1 m 1bmmod m 1a ma1am 1Mm1 1 mbm 1mod a 1a1a m1 Nm 1 m 1M 1 1 are fulfilled where integers M 1 1 are defined consequently from the equalities a b M1 a 1a1a b M 1 M 1 1 m1 ma m 1a1 am bm Mm 1 m 1 Mm 1 1 m m 1a ma1am 1 bm 1 Mm 1 m Mm m 1 a 1a1 a m 1 Nm 1 m 1 b M M m 1 1 1 1 b mod m Proof The roof of the Theorem can be obtained by similar way to the roofs of Theorems 1 and Eamining various cases of a and b we need to study only the case a > and b Because of aearance of uncertainty of a solution we divide this case to a >1 and a 1 Theorem 4 Let a b and b b1 1 or Then to be a solution of Equation (1) in if and only if he net congruences bmod b b1mod9 1a M1bmod P 1a 1 a 1 1 M 1bmod 1 1P 1 1 a a1a M 1 1 b mod 4 are fulfilled where integers M 1 1 are defined from the equalities b b1m19 1a b M1 M 1 P 1a 1 a11 b M 1 M 1 1 1P 1 1 a a1a b M M 4 1 1 1 1 Proof Analogously to the roof of Theorem 1 Theorem 5 Let a b 1 1 Equation (1) in if and only if he net congruences and b b or Then to be a solution of the mod9 mod mod b mod b b M b P a M b P a a a M b mod 4 1 1 1 1 1 1 1 1 1 1 4 1 1 1
I M RIKHSIBOEV ET AL 41 are fulfilled where integers M 1 1 are defined from the equalities 9 4 b b1 M1 bm M 1 1 1 P 1 a 1 b M 1 M 1 P a a a b M M 1 1 1 1 4 1 1 1 1 1 Proof Analogously to the roof of Theorem 1 Similarly to Theorem 4 it is roved the following Theorem 6 Let a m 4 b Equation (1) in if and only if he net congruences and b b 1 or 1 Then to be a solution of the bmod b b1mod9 1M1bmod P 1 1 M 1bmod 1 1P 1 1 b mod 4 m1 m1 m1pm 1 m 1 m a bmmod 1 1 P 1 1 ma am b mod m1 are fulfilled where integers M 1 1 are defined from the equalities b b1m19 1 b M1 M 1 P 1 1 b M 1 M 1 1 1P 1 1 b M 1 1 M 1 14 m1 m1 m1pm m 1 m a bm Mm 1 m Mm m 1 1 1P 1 1 ma a m b M M m1 1 1 1 1 Now we consider Equality () with a 1 b Put A A a A a R where R 1 1 i N P 1 1 s1 s1 i N 1 i P1 N s S s i N 1 1 P 11 s1 s1 Theorem 7 Let a 1 b and solution of Equation (1) in if and only if the congruences to be so that A a mod Then to be a
4 I M RIKHSIBOEV ET AL b 1 1 a 1a1a1N M mod a M b b mod mod are faithfully where M and integers M 1 a M bm 1 1 b are defined from the equalities a 1N 1 a1a 1M b M 1 1 b mod 1 mod has a solution Then denote by M a M1 b1 Using Theorem 1 we have eistence of solutions of the congruences Proof Let the congruences b a b M the number satisfying the equality a 1a1a1N M b mod The net chain of equalities 1 1 1 1 a a N a a b M b M M b M M 1 1 1 1 1 1 1 b b 1 shows that is a solution of Equation (1) From the roof of Theorem 7 it is easy to see that if A a mod then we have the following congruences and aroriate equalities a) b mod ie b M1 ; b) a M 1 b 1 mod then a M1 b1m ; c) a 1M b mod then am b M (4) 1 ; A 1 a 1 1 a mod 1 1 M b it follows that d) Since A 1 a 1 1 a 1 1 M b M 4 1 A A1 a1 1 then the congruence d) can be written in the form A a M b 1 1 1 1 mod and so we have A a M b M 1 1 1 1 4 1
I M RIKHSIBOEV ET AL 4 If for any natural number we have mod Equation (1) However if there eists such that A then we could establish the criteria of solvability for A mod then the criteria of solvability can be found and therefore we need the following Lemma 1 Let a 1 b and to be so that A mod1 A mod for some fied If be a solution of Equation (1) then it is true the following system of the congruences b 1 1 1a a 1a 1S M 1 1b mod A 1a1a a S 1M 1 1 1 1 mod a M b b where 1 mod mod i A i i 1a1ia a i S 1i M 1 i 1 i 1 b 1 imod and integers M are defined from the equalities M1 b M a M b 1 1 S 1M 1 1 1b 1 M a a S M b 1 1 1 1 1 M A a a a 1 1 1 i 1 i M A a a a i i1 1 i i S M b i 1 i 1 i 1 i 1 1 i Proof We will rove Theorem by induction Let 1 ie A A a mod A1 a11 mod 1 1 then the system of the congruences (4) are true Note that S S 1 From () it is easy to get A t t t a 1 t a a t 1 S t 1 t M t t b t mod t 4 Therefore A a a S M b mod t4 (7) t 1 t t t1 t t t t (5) (6) where t M A a a S M b t 4 t1 t 1 t t t1 t t t t Obviously the statement of Lemma is true for 1 ie for i t A1 Let ie A mod A1 mod A a 1 mod then from the equa- lities (7) it follows that the following congruences are be added to the system (4): a a S M b mod it follows e) 1 4 4 1 4
44 I M RIKHSIBOEV ET AL M a a S M b ; 5 1 1 4 4 1 4 A a a 1 a 4 S 5 M 5 1 b 5 mod it follows 1 f) A1 M6 1 a a 1 a4 S5 M5 1 b5; A h a a a S M b where 5 M are defined by equalities 1 t1 ) t t t t t1 t1 1 t t1 mod t and t 1 t A1 t1 Mt 1 t t ta ta at St 1 Mt 1 1 tbt 1 Since S4 S S t1 t 5 1 St 1 St 1 1t t we denote by i t 4 and have e) a 1 a M4 1 b4 mod A a a M b mod f) 1 4 5 1 5 i h) A a a a S M b mod where i i1 i 4 i4 i5 i5 1 i1 i5 i M5 1 a 1 a M4 1 b4 M6 1 A a 1 a4 M5 1 b5 M A a a S M b 6i i i i1 i4 i5 i5 i1 i5 So we showed that the statement of Lemma is true for Let the system of congruences (5) and (6) is true for Since mod A then from the congruences A a a a S M b mod i1 i i i 1 1 i i 1 i 1 i 1 i 1 1 i we derive a a a S M b mod 1 1 1 1 A 1 1 a 1 a a S M b mod A i 1 i 1 i a 1 i a a i S i M i 1 i b i mod i 1 It is easy to chec that S S R S S R i 1 1 i i 1 1 1 1 i i 1 1i By these correlations we deduce For i 1 we get Consequently we have A a a a S M 1 1 1 1 1 A a a S M 1 1 1 1 A i 1 i 1 i a 1 a i S i M i 1 i A a a S M i 1 1 1i i i i i 1 i
I M RIKHSIBOEV ET AL 45 a a a S M b mod 1 1 1 1 1 1 1 1 A1 11 a a S M b mod i 1 A a a S M b mod i 1 1 1i i i i i 1 i i So we established that the system of congruences (5)-(6) is true for 1 Using the Lemma 1 we obtain the following Theorems Theorem 8 Let a 1 b and to be such that A mod1 A mod for some fied 1 Then to be a solution of the Equation (1) in if and only if the system of the congruences bmod a M1b1mod 1a a 1a 1S M 1 1b mod A a a a S M b mod 1 1 1 1 1 1 1 has a solution where 1 and integers M 1 1 M1 b M a M b 1 1 are defined from the equalities M 1 M a a a S M b 1 1 1 1 1 1 A a a a S M b 1 1 1 1 1 1 1 Theorem 9 Let a 1 b and A mod for all Then to be a solution of the Equation (1) in if and only if the system of the congruences bmod a M1b1mod 1a a 1a 1S M 1 1b mod A 1a1a a S 1M 1 1 1 b mod 1 has a solution where 1 and integers M 1 1 M1 b M a M b 1 1 to be so that are defined from the equalities M a a a S M b 1 1 1 1 1 1 M A a a a S M b 1 1 1 1 1 1 1 1 Acnowledgements The first author was suorted by grant UniKL/IRPS/str1161 Universiti Kuala Lumur REFERENCES [1] F M Muhamedov B A Omirov M Kh Saburov and K K Masutova Solvability of Cubic Equations in -Adic Integers > Siberian Mathematical Journal Vol 54 No 1 51-516
46 I M RIKHSIBOEV ET AL [] K Hensel Untersuchung der Fundamentalgleichung Einer Gattung fur Eine Reelle Primzahl als Modul und Besrimmung der Theiler Ihrer Discriminante Journal Für Die Reine und Angewandte Mathemati Vol 11 No 1 1894 61-8 [] A A Buhshtab Theory of Numbers Moscow 1966 84 [4] S B Kato -Adic Analysis Comared with Real MASS Selecta 4 [5] N Koblitz -Adic Numbers -Adic Analysis and Zeta-Functions Sringer-Verlag New Yor Heidelberg Berlin 1977 19 htt://ddoiorg/117/978-1-4684-47- [6] V S Vladimirov I V Volovich and I Zelenov -Adic Analysis and Mathematical Physics World Scientific Singaore City 1994 htt://ddoiorg/1114/1581 [7] S Albeverio Sh A Ayuov B A Omirov and A Kh Khudoyberdiyev n-dimensional Filiform Leibniz Algebras of Length (n 1) and Their Derivations Journal of Algebra Vol 19 No 6 8 471-488 htt://ddoiorg/1116/algebra7114 [8] Sh A Ayuov and B A Omirov On Some Classes of Nilotent Leibniz Algebras Siberian Mathematical Journal Vol 4 No 1 1 18-9 htt://ddoiorg/11/a:148914 [9] B A Omirov and I S Rahimov On Lie-Lie Comle Filiform Leibniz Algebras Bulletin of the Australian Mathematical Society Vol 79 No 9 91-44 htt://ddoiorg/1117/s497791x [1] I S Rahimov and S K Said Husain On Isomorhism Classes and Invariants of Low Dimensional Comle Filiform Leibniz Algebras Linear and Multilinear Algebra Vol 59 No 11 5- htt://ddoiorg/118/818957646 [11] Sh A Ayuov and T K Kurbanbaev The Classification of 4-Dimensional -Adic Filiform Leibniz Algebras TWMS Journal of Pure and Alied Mathematics Vol 1 No 1 155-16 [1] A Kh Khudoyberdiyev T K Kurbanbaev and B A Omirov Classification of Three-Dimensional Solvable -Adic Leibniz Algebras -Adic Numbers Ultrametric Analysis and Alications Vol No 1 7-1 [1] M Ladra B A Omirov and U A Roziov Classification of -Adic 6-Dimensional Filiform Leibniz Algebras by Solution of q a Central Euroean Journal of Mathematics Vol 11 No 6 1 18-19 htt://ddoiorg/1478/s115-1-5-9 a in the Field of -Adic Num- [14] J M Casas B A Omirov and U A Roziov Solvability Criteria for the Equation bers 11 arxiv:11156v1 [15] F M Muhamedov and M Kh Saburov On Equation 1 55-58 [16] T K Kurbanbaev and K K Masutova On the Solvability Criterion of the Equation from Uzbe Mathematical Journal No 4 11 96-1 q q a over Journal of Number Theory Vol 1 No 1 a b in with Coefficients