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Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by…

We give two elementary proofs, at a level understandable by students with only pre-calculus knowledge of Algebra, of the well known fact that an irreducible irrational n-th root of a positive rational number cannot be solution of a…

This article introduces an intuitive function MY that simplifies solving cubic equations without venturing into the complex space. To many, it's quite strange that cubic root(s) are expressed using trigonometric functions in the…

It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…

An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.

We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given…

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…

The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not…

We establish a criterion for a complex number to be algebraic over Q of degree at most two. It requires that, for any sufficiently large real number X, there exists a non-zero polynomial with integral coefficients, of degree at most two and…

Starting from the solution to Bring's equation the root ambiguity is removed from the solution to the quintic equation. This gives the five complex roots of the quintic equation as indicated by Gauss's Fundamental Theorem of Algebra.r

In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…

We show that a polynomial equation of degree less than 5 and with real parameters can be solved by regarding the variable in which the polynomial depends as a complex variable. For do it so, we only have to separate the real and imaginary…

We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that…

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

This paper's central theme is to prove the existence of an n-algebra whose multiplication cannot be expressed employing any binary operation. Furthermore, to prove if two algebras are not isomorphic, this property does not hold for…

This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5 + mt^3 + nt^2 + pt +…

A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…

In this study, new master theorems and general formulas of integrals are presented and implemented to solve some complicated applications in different fields of science. The proposed theorems are considered to be generators of new problems,…

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or…