Rationalizing Denominators Using Gröbner Bases

The problem of rationalizing denominators for two types of fractions is discussed in the paper. By using the theory and algorithms of Gröbner bases, we first introduce a method to rationalize the denominators of fractions with square root and cube root, and then, for the denominators with higher radical of the general form, the problem of rationalizing denominators is converted into the related problem of finding the minimal polynomials. Some interesting results and an executable algorithm for rationalizing the denominator of these type fractions are presented. Furthermore, an example is also established to illustrate the effectiveness of the algorithm.

Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two

Suppose that p and q are two distinct odd prime numbers with n = p q . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤ n was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over F l was presented and their minimal polynomials and linear complexities were derived, where l = r s with a prime number r and gcd l , n = 1 . The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over F l with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a 0 , . . ., an− 1, where n is the degree of the minimal polynomial of a over F .

The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

Denote the alternating and symmetric groups of degree [Formula: see text] by [Formula: see text] and [Formula: see text], respectively. Consider a permutation [Formula: see text], all of whose nontrivial cycles are of the same length. We find the minimal polynomials of [Formula: see text] in the ordinary irreducible representations of [Formula: see text] and [Formula: see text].

Automated Detection of Interesting Properties in Regular Polygons

We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of $$\cos \frac{2\pi }{n}$$ cos 2 π n , and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.