In Mathematics, the theory of equations comprises a major part of traditional algebra. Topics range from polynomials, algebraic equations, separation of roots, including Sturm’s theorem, approximation of roots, and the application of matrices and determinants, to the solving of equations. From the point of view of abstract algebra, the material in the theory of equations is divided between symmetric function theory, field theory, Galois theory, and computational considerations, including numerical analysis.
The subject matter of the book is divided into 9 chapters. Chapter 1, “General Properties of Equations”, describes the equations of higher degrees and investigates similar relations which hold in the case of equations of the nth degree. Some elementary properties of equations have been studied. Chapter 2, “Relations between the Roots and Coefficients of a Polynomial Equation of the nth Degree”, explains how to establish the relations between roots and coefficients of a polynomial equation of the nth degree. Chapter 3 deals with the symmetric as well as asymmetric functions of the roots. Chapter 4 explains the methods of transformation of equations.
In Mathematics, Descartes’ rule of signs is a technique for determining the number of positive or negative real roots of a polynomial. The rule gives us an upper bound number of positive or negative roots of a polynomial. It is not a complete criterion, i.e., it does not tell the exact number of positive or negative roots. The same has been described in chapter 5, with both the positive roots and negative roots. Chapter 6 explains Cardano’s method of solving cubic equations in detail, while solutions of biquadratic equations have been studied in chapter 7.
Sturm’s sequence of a polynomial and Sturm’s theorem have been explained in chapter 8. Solutions of numerical problems have been given in chapter 9.
Equations and theorems have been explained in simple language and lucid manner for easy understanding by the readers. Illustrative examples have also been given. There are exercises in all the chapters to give the students a feel of the type of questions they should expect in the examination. Answers to all the exercises have been provided. The book will be very useful for the students of Mathematics at graduation level. It will also help those preparing for competitive examinations involving mathematical problems.
Hari Kishan is a Senior Reader in Mathematics at Kishori Raman Postgraduate College, Mathura, affiliated to Dr. Bhimrao Ambedkar University, Agra, with over thirty years of teaching experience. He completed his M.Sc. from B.S.A. College, Mathura, in 1971, obtaining a record percentage of marks for which he was awarded Gold Medal by Agra University. He received a Ph.D. in Mathematics (Fluid Dynamics) in 1981 from the same university for his research on “Flow of Homogeneous or Stratified Viscous Fluids”. He is a prolific author and has published numerous research papers in reputable mathematical journals as well as several books, including Trigonometry, Integral Calculus, Coordinate Geometry of Two Dimensions, Matrices, Modern Algebra, Theory of Equations, Dynamics, Statics, Real Analysis, Numerical Analysis and Sure Success in Convergence.
