QUADRATICINT - A MAPLE PACKAGE FOR WORKING WITH
ELEMENTS OF O. O. Trebenko, N. M. Tsybulska

(National Pedagogical Dragomanov University, Kyiv, Ukraine)

March 18, 2016

Abstract

This paper presents the description of the package for working with elements of    where   is a squarefree integer. Given in the paper are the program codes of package commands and examples. Developed software will be useful for specialists in the field of abstract algebra and applications.

Key words: Computer algebra, symbolic computation, algebraic number theory, ring theory, rings of integers of algebraic number fields, rings  .

Mathematics Subject Classification (2010): 11R04, 11R11, 11Y40, 11Y70

1. INTRODUCTION

Recently, attention to research of the rings  rapidly increases. Primarily, this is due to the search for the solution of many unresolved problems raised within the algebraic theory, which today is extremely important to modern mathematics. In addition, there is an applied interest for algebraic numbers have a broad application, especially in cryptography (code Hill) and topology.

In applications of algebraic numbers (especially in cryptography) it is more often needed to decompose numbers into the prime factors not in rings  , but in  . Note that by   we mean the minimal ring containing  and the element  , namely the ring . The ring    coincides with the ring   when   and is its proper subring when  .

Due to the above, modern theoretical research in algebraic number theory and applications need a special package/module for working with elements of   . But, unfortunately, existing SCM don’t contain such package.  

As a result of investigations of quadratic number rings properties [1]-[4] authors have developed some procedures that could be useful for the following investigation and for applications.  

The authors created such a package for SCM Maple. This paper presents the characterization of the package, gives the program codes and examples.

2. THE STRUCTURE OF THE PACKAGE QUADRATICINT

Calling Sequence

QuadraticInt[command](arguments)

command(arguments)

Description

•           The QuadraticInt package is a collection of commands for working with elements of the principal ideal ring  , where  is a squarefree integer.

•           Note that by   we mean the minimal ring containing Z and the element  , namely the ring  . The ring   coincides with the ring   when   and is its proper subring when  .

List of QuadraticInt Package Commands

Let   and let   be a ring of principal ideals. The following is a list of available commands:

1. SGIiselementof(a,d): determines whether the element belongs to some ring   and returns  .

2. SGInorm(a,d): returns the norm of the element    in  .

3. SGInormal(x,d): returns the element    if  and returns the element    if  (is used to provide the uniqueness of the result in the following procedures).

4. SGIquo(y1,y2,d,’r’): returns the quotient of   divided by  in  . The third argument is optional. If it is included in the calling sequence for SGIquo, it will be assigned the remainder  .

5. SGIrem(a,b,d,’q’): returns remainder of   divided by  in  . If a third argument is included in the calling sequence for SGIrem, it will be assigned the quotient q.

6. SGIgcd(a,d): computes the greatest common divisor of an arbitrary number of elements in  .

7. SGIlcm(a,d): computes the least common multiply of an arbitrary number of elements.

8. SGIlrgcd(a1,b1,d): returns  q= SGIgcd({a1,b1}, d), and optionally  such that  .

9. SGIisprime(n,d): is a primality testing routine. It returns true if    is prime in  . It returns false otherwise.

10. SGIfactor(n,d): factorization of    in  .

11. SGIfacset(n,d): computes the set of prime factors of the element  .

12. SGIfactors(n,d): factorization is returned in the form    where  ,   – prime elements,   – their multiplicities,    – some unit.

13. SGIdivisor(a,d): computes the set of all divisors of   (up to associates).

14. SGIphi(u,d): calculates Euler function for  .

15. SGIorder(n,m,d): order of    modulo  .

16. SGIbasis(x,y,d): determines whether the elements  ,    form a basis of vector space  .

17. SGIissqr(n,d): test if    in    is a perfect square.

18. SGIareassociated(a,b,d): test if    and    are associates in  .

3. DESCRIPTION OF QUADRATINT PACKAGE PROCEDURES

4. Conclusions

Proposed in the paper is a package for working with elements of  in SCM Maple. It will be useful both for further investigations and for practical applications.

5. References

[1] Тrebenko О.O, Tsybulska N.M. Prime elements of rings   // Nauckovyi chasopys Natsionalnogo Pedagogichnogo Universitetu imeni M.P. Dragomanova (Scientific journal of the National Pedagogical Dragomanov University). Series 1. Physical and Mathematical Sciences: Collection of Scientific Papers. – Kyiv: Dragomanov NPU, 2014. – №16 (1). – P. 219-227.

[2] Tsybulska N.M. Rings   : properties and practical applications: student research work. – Kyiv: 2016. – 53 p. – Bibliogr.: p. 29-30. (In Ukrainian)

  [3] Tsybulska N.M. Rings   : properties and practical applications // Education and science – 2016. Collection of scientific papers. – Kyiv: Dragomanov NPU, 2016. – P. 243-244. (In Ukrainian)  

  [4] Tsybulska N.M. The Euler   function on the set   // Proceeding of the Fifth scientific conference of young scientists on mathematics and physics «Actual problems of modern mathematics and physics and methods of their learning» (April 25-26, 2016, Kyiv, Ukraine). – Kyiv: Dragomanov NPU, 2016. – P.59. (In Ukrainian)