Technical Books

This PDF covers the following topics related to Set Theory, General considerations, Basic concepts, Constructions in set theory, Relations and functions, Number systems and set theory, Infinite constructions in set theory, The Axiom of Choice and related properties, Set theory as a foundation for mathematics.

This PDF covers the following topics related to Set Theory, Introduction, Wellorders and Ordinals, Classes and Transfinite Recursion, Cardinals, Zorns Lemma, Ramseys Theorem, Loss Theorem, Cumulative Hierarchy, Relativization, Measurable Cardinals, Godels Constructible Universe, BanachTarski Paradox.

This PDF covers the following topics related to Set Theory and Forcing, Introduction, Axioms of Set Theory, Class Relations, Functions, Families of Sets and Cartesian Products, Ordinals and Cardinals, Classes and Sets, WellOrderings and Ordinals, Inductive Definitions, Stratified or Ranked Classes, Ordinal Arithmetic, Cardinals and Their Arithmetic, Foundation, Relativization, Consistency of the Axiom of Foundation, Inaccessible Ordinals and Models of ZFC, The Reflection Scheme, Formalizing Logic in U, Model Theory for Uformulas, Ordinal Definability and Inner Models of ZFC, The Principle of Choice, Constructibility, Formulas and Absoluteness, The Generalized Continuum Hypothesis in L, Forcing, Generic Extensions, Mostowski Collpase of a Wellfounded Relation, Construction of Generic Extensions, Definition of Forcing, etc.

This note describes the following topics, Propositional calculus, Well orderings and ordinals, Posets and Zorns lemma, Predicate logic, Set theory, Cardinals and incompleteness.

This note contains below topics, Origins of Proof Theory, Flavours of Proof Theory, The Logics, Hilbert Systems, Natural Deduction, Sequent Calculi.

This book contains below topics, and discuss how Boolean ultr apowers can be applied in urelement set theory, ZFCU R Collection A, Over ZFUR, for every cardinal, the following are equivalent, DC scheme, If duplication holds over A A, then homogeneity holds over A, Assume Tail and AC A Duplication holds over some set of urelements, Over ZFCUR, the following implications hold, DC Ord Collection, RP Collection.

These notes are meant as an introduction to descriptive set theor y. This book contains below topics, Borel Sets and the Borel Hierarchy, Analytic Sets, Reductions, Uniformization.

This note covers the following topics, Introduction to number rings, Ideal arithmetic, Explicit ideal factorization, Linear algebra for number rings, Geometry of numbers, Zeta functions, Computing units and class groups, Galois theory for number fields.

Aim of this book is to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. Topics covered includes, Sets, Integers, Functions, Groups, Rings and Fields.

This wikibook explains ring theory. Topics covered includes, Rings, Properties of rings, Integral domains and Fields, Subrings, Idempotent and Nilpotent elements, Characteristic of a ring, Ideals in a ring, Simple ring, Homomorphisms, Principal Ideal Domains, Euclidean domains, Polynomial rings, Unique Factorization domain, Extension fields.

On the one hand this book intends to provide an introduction to module theory and the related part of ring theory. Topics covered includes, Elementary properties of rings, Module categories, Modules characterized by the Homfunctor, Notions derived from simple modules, Finiteness conditions in modules, Dual finiteness conditions, Pure sequences and derived notions, Relations between functors and Functor rings.

This PDF covers the following topics related to Rings and Fields, A brief overview, An introduction to Rings, Integral Domains and Fields, Homorphisms, ideals and quotient rings, Prime ideals, maximal ideals, and fields of quotients, Euclidean Domains, Factorisation in polynomial rings, Vector spaces, Extension fields, Straightedge and Compasses constructions.

This PDF covers the following topics related to Rings and Fields, A pseudohistorical note, More on subrings and ring homomorphisms, The evaluation or the substitution map, Defining fractions, Using the universal property of the field of fractions, An application of the first isomorphism theorem, The factor theorem and the generalized factor theorems, Gaussian integers, Irreducibility and zeros of polynomials, Content of a polynomial with rational coefficients, An example on the mod irreducibility criterion, Factorization, uniqueness, and prime elements, Ring of integer polynomials is a UFD, Greatest common divisor for UFDs, Extension of isomorphisms to splitting fields, Finite fields, etc.

This PDF covers the following topics related to Groups, Rings and Fields , Familiar algebraic systems, review and a look ahead, Binary operations, and a first look at groups, Interlude, properties of the natural numbers, Integers, Polynomials, Equivalence relations, and modular arithmetic.

This note explains the following topics, Numbers, Definitions and first examples, Further axioms for rings, Subrings, Ideals, Quotients of rings, Homomorphisms, Three isomorphism theorems, Prime ideals, Maximal ideals, Divisors, Irreducibles and prime, Euclidean rings, Greatest common divisors again, Some nasty examples, The classification of finite field.