Ramji Lal

Field Theory, Galois Theory

This chapter is devoted to the theory of fields, Galois theory, geometric constructions by ruler and compass, and the theorem of Abel–Ruffini about the polynomial equations of degree \(n, n\ge 5\). We also discuss cubic and biquadratic equations.

Arithmetic in Rings

This chapter is devoted to the study of rings in relation to their arithmetical properties.

Canonical Forms, Jordan and Rational Forms

This chapter is devoted to describe the similarity classes with entries in a field. For this purpose, we introduce the concept of modules and obtain the structure of finitely generated modules over a principal ideal domain.

Structure Theory Continued

This chapter deals with the Remak-Krull-Schmidt Theorem on direct decomposition, structure theory of solvable and nilpotent groups together with the presentation theory of groups.

Language of Mathematics 2 (Set Theory)

This chapter contains a brief introduction to set theory which is essential for doing mathematics. There are two main axiomatic systems to introduce sets, viz. Zermelo–Fraenkel axiomatic system and the Godel–Bernays axiomatic system. We follow the Zermelo-Fraenkel axiomatic system together with the axiom of choice.

Permutation Groups and Classical Groups

The two main sources of groups are the permutation groups and the matrix groups. This chapter is devoted to introduce these groups, and to study some of their fundamental and elementary properties.

Language of Mathematics 1 (Logic)

The principal aim of this small and brief chapter is to provide a logical foundation to sound mathematical reasoning and also to understand adequately the notion of a mathematical proof. Indeed, the incidence of paradoxes (Russell’s and Cantor’s paradoxes) during the turn of the nineteenth century led to a strong desire among the mathematicians to have a rigorous foundation to all disciplines in mathematics. In logic, the interest is in the form rather than the content of the statements.

General Linear Algebra

The present chapter is devoted to the study of Noetherian rings, Projective modules, Injective Modules, Tensor product of modules, Grothendieck, and Whitehead groups of rings.

Structure Theory of Groups

The main problem in the theory of groups is to classify them, and to study their structures. The present chapter deals with the Sylow theorems which play a very crucial role in the structure theory of groups. This chapter also contains the classification of finite abelian groups together with the Schreier and the Jordon-Holder theorems.

Elementary Theory of Rings and Fields

Ring is an important algebraic structure with two compatible binary operations whose intrinsic presence in almost every discipline of mathematics is frequently noticed. The theory of rings, in the beginning, will be developed on the pattern the theory of groups was developed.