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Extension field is the most fundamental concept in Algebraic Number Theory. It plays a vital role while examining the roots of a polynomial equations through Galois Theory. Moreover field extensions are widely used in Algebraic Geometry. This article depicts FIELD EXTENSION THEORY and ALGEBRAIC EXTENSIONS in a recreational way. As this article is written in the current year 2023 and might be published in the year 2024 we have chosen those year's numbers 2023 and 2024 and made an attempt to demonstrate the central ideas and fundamental concepts of Galois Theory in an elegant apprach. [ABSTRACT FROM AUTHOR]

In this research article we proposed some important applications of Matrix Algebra tools. Role of determinants in fitting a regression line has been the presented here. Moreover, how the matrices are useful in Information theory is also discussed. The importance of Modal Matrix in obtaining the general solution of first order SLDE is shown. Lastly the use of determinants in Analytical Geometry are also discussed. [ABSTRACT FROM AUTHOR]

The research article explores on applications of Linear Algebra in various branches of science namely Space Geometry, Chemical Sciences, Optimization and Cryptography. Though one is familiar with the concepts of Linear Algebra, real-world applications of it should be understood. In this view the authors are believing that they are successful in presenting some wonderful applications of Linear Algebra. [ABSTRACT FROM AUTHOR]

Fixed power sum of positive integers is a well-known problem in number theory. Here we have given a brief history about it by mentioning the contributions of some eminent mathematicians. An internal relation between Σλ and Σλ3 has been traced in an geometric approach and the authors are strongly believing that this geometric approach can give a generalized result for Σλn. Pascal's recurrence relation has been proposed with illustrative examples. Moreover a recurrence relationship is presented by the authors using binomial theorem and they are concluding that any fixed power sum can be easily carried out with this relationship. More importantly by applying binomial theorem we have proved that any fixed power sum is a unique polynomial over rational numbers. Finally an explicit formula has been proposed and this can give any power sum formula. [ABSTRACT FROM AUTHOR]

SVD is a most powerful concept extensively used in a large number of models. It plays an important role in PCA. As it describes the factorization of matrix, it is also used in matrix completion. Hence we proposed the concept of SVD here with an illustrative example. If a linear system has no solution, the pseudo-inverse will help us in finding the BEST-FIT solution. In new of this, we (the authors) presented the mathematical concept behind it and shown one illustrative example. [ABSTRACT FROM AUTHOR]

Objective of this writing is to present excellent applications of the theory of matrices in various fields. We have given some illustrative examples of uses of matrix algebra in Cryptography, Vector mechanics, Linear dependence of functions and Chemical Sciences. It is fair to say that almost all branches of STEM(Science-Technology-Engineering-Mathematics) are widely using the matrix algebra tools. Hence in this article we(the authors) have made an attempt to bring them to the notice of young researchers. [ABSTRACT FROM AUTHOR]

In this paper we have presented various generalized results on power summation with proofs and examples. All these results are given is recurrence relations so that one can get any fixed power sum starting with σ n Backward Difference operator is employed in order to get the power sum. A lemma, we proposed here, is a significant one as it can give σ n,σ n2, σ n3, .... Moreover we have discussed the applications of Binomial Coefficients in computing the desired result. [ABSTRACT FROM AUTHOR]

The main objective of this article is to discuss the literature on the sum of powers of integers and propose some generalized results with interesting numerical examples. This power sum is evaluated with one recurrence relation based on ar eas of rectangles. We have described the proof of the recurrence relation with the universal rectangle and its interior rectangles. In this paper Pascal's generalized result is given from which power sum for any fixed positive integer can be derived. Illustrat ive examples in Pascal's formula will clear the methods of computing the sums namely sum of first n-natural numbers, sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers and so on. We have given a powerful formula for power sum using Bernoulli's coefficients. First ten Bernoulli's coefficients are calculated and are used in extracting first ten power sums. [ABSTRACT FROM AUTHOR]

This paper explores on deriving the formulae for fixed power sum of natural numbers. A review of literature on power sum is presented in a lucid manner. Here a general method is depicted by applying some results in differential calculus. Moreover, an important application of Pascal's triangle in computing this sum is discussed. Finding the generalized formula for the power sum has been a fascinating problem for centuries in the research world of Analytical Number Theory. Following discussion presents number of interesting results on this sum with illustrative examples. At first, we gave the early history of the power sum and then move to the various methods of finding this sum. We have proposed a double series formula to the power sum from which the power sums upto 10th order is calculated. [ABSTRACT FROM AUTHOR]