SIMULATING ALGEBRAIC EXTENSIONS OF FINITE FIELDS INTERACTIVELY USING MAPLET AS A STRATEGY OF LEARNING ABSTRACT ALGEBRA

Extended abstract\ud \ud 1. Backgrounds, Objectives and Learning Strategy\ud Mathematics is a (written) language (see [6]). Unlike the ordinary and computer programming languages, however, mathematics has very different objects to be communicated. The objects in mathematics as a language belong to every mind of human beings that cannot be physically sensed but can be symbolically written. \ud With objectives to revitalize and to disseminate mathematical concepts that need various written mathematical symbols, an application that can display and simulate interactively and visually any input-output processes is needed. In this paper, Maplet, an application available in the computer algebra system Maple, is used. Since the application can visually display outputs of advanced mathematical computations interactively according to some inputs given by its users, it can be used as a good learning tool in mathematics. Any interactive visual presentation of mathematical concepts and applications is a learning strategy that agrees with the paradigm of SCL (student-centered learning). \ud This paper will discuss the usages of Maplet to represent some theories of finite fields with characteristic 2, e.g. algebraic field extension based on a primitive root of a primitive polynomial p(x) ??? GF(2)[x] of degree m, the splitting-field that contains all roots of xN ??? 1 where N = 2m ??? 1. These mathematical concepts are fundamental ingredients of cyclic codes in the theory of error-correcting codes. The cyclic codes constitute an important class of linear codes that has many applications in digital communications and in the store-retrieve methods of digital data.\ud The interactive visual representation with Maplet will familiarize the learners with examples of m-dimensional vector space over GF(2) which can be identified with the finite field GF(2m) and ultimately with many more algebraic entities like primitive and minimal polynomials, the splitting fields of xN ??? 1 and other fundamental concepts of binary cyclic codes before the learners were given the formal theories of those algebraic concepts.\ud The above learning strategy, which would be called familiarization sooner, theory later, is implicitly proposed and recommended throughout this paper. This learning strategy is more recommended for more difficult concepts. Therefore, the most appropriate target of learners is the adolescene-adulthood learners as one of cognitive stages proposed by Piaget and the strategy is similar to the Piaget???s concept of discovery learning [1, p.102]. \ud From author???s classroom experiences, abstract algebra or a related topic in mathematics (e.g. error-correcting codes) is one of the most difficult concepts to be comprehended by students despite the fact that there are abundance of good algebraic error-correcting textbooks for students, e.g. the classical textbook [3] by MacWilliams and Sloane, or the later and popular textbooks [2] by Huffman and Pless or more recent textbooks like [5] by Ryan and Lin. \ud 2. Using Maplets for Learning Abstract Algebra\ud The dynamics of input-output data of the created Maplets is exploited to give the learners illustrations how an algebraic concept is developed or derived from other concepts. There are many Maplets created by author with different topics (cryptography, cyclic groups, combinatorial games, etc) and with different learning objectives. However, this paper will discuss only one particular Maplet that illustrates a contruction of m dimensional vector space F2m and its equivalence field: the Galois field GF(2m), when the given Maplet input is a primitive polynomial of degree m. The same Maplet can also be used to illustrate the factorization of xN ??? 1 into linear factors, so it can be used to illustrate the derivation of splitting field.\ud The first box of the Maplet should be filled in with a primitive polynomial p(x) of a chosen degree m. After clicking a button to verify the primitiveness of p(x), the user can continue to click the next button and a Maplet window displays all elements of the associate field GF(2m) as polynomials with indeterminate a of degree less than or equal to m. The a is a zero of p(x) and indeed a primitive element of GF(2m). \ud The next Maplet box receives (products of) polynomials in GF(2m)[x] as an input and the next Maplet window displays the output of the input as ordinary product of polynomials where the rule of the polynomial products supposedly has been taught in high school. The last Maplet window displays the same output but treated as an element of GF(2m)[x]/<xN ??? 1>. In particular, when the given polynomial in the last input box is a single power of a, say an, the last Maplet windows will display its remainder when divided by p(a). When a product of high degree polynomials (with indeterminate x) is given, the Maplet will decrease the order of the polynomial to its remainder when divides by xN ??? 1. When the output is a polynomial in GF(2)[x]/<xN ??? 1>, it must be a (products of) minimal polynomial. This result will lead the instructor to introduce new concept: cyclotomic cosets, only by examining the exponents of the indeterminate x. \ud The following tables gives some examples of inputting Maplet with (product of) polynomial(s) as input and their output when running the Maplet. \ud \ud Table 1. Learning Roles of Cyclotomic Cosets with Maplet\ud \ud Note that a Maplet user can change the values of m or even the primitive polynomial from which the finite field GF(2m) is derived. \ud The same Maplet can be used for different objectives. The instructor only needs to slightly change the inputs as the following table illustrates it.\ud Table 2. Learning Generator and Parity-Check Matrices with Maplet\ud \ud 3. Conclusions\ud \ud The uniqueness of mathematics, especially abstract algebra, as a (written) language, warrants a suited learning methods and strategy for studying mathematics. This paper recommends a strategy that familiarize a mathematical subject first through related examples and concepts interactively using Maplets before the learners get the formal concepts of the subject. This paper discuss a particular Maplet as an example how the Maplet can be used for different objectives in the theory of finite fields