Theory of Equations
"Art of mathematics consists finding the special case which contains all germs of generality." -David Hilbert
In mathematics, the Theory of Equations comprises a major part of traditional algebra. Topics include 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 is divided between symmetric function theory, field theory, Galois theory, and computational considerations including numerical analysis.
The first chapter Systems of Equations is basically about Linear Algebra, that is the study of multivariable equation systems through matrices and vector spaces. It extends to complete study Matrices and related topics of Determinants, Eigenvalues,etc. along with their applications in equation systems.
The second chapter Polynomial Functions is about the study of polynomials and their properties such as Roots and Determinants, Coefficients and Symmetric Function of Roots, Galois Groups, Derivatives, Maxima and Minima, etc.
The third chapter Functional Equations