MM/AM-201 SEMESTER – II                   Paper-I : Galois Theory

Course Objectives:

 The course aims to: 

1.Introduce the theory of field extensions and irreducible polynomials.

2. Develop a strong understanding of splitting fields, normal and separable extensions.

3. Explain the structure of automorphism groups and their role in Galois theory.

4. Apply Galois theory to classical algebraic problems such as solvability by radicals and ruler and compass constructions.

Course Outcomes: 

On successful completion of the course, students will be able to

1. Identify and construct algebraic extensions of fields and analyze irreducible polynomials.

2. Understand and apply the concepts of splitting fields, normal extensions and separability.

3. Describe and determine automorphism groups and fixed fields.

4. Apply the concepts of Galois theory to analyze and solve classical algebraic problems such as cyclotomic polynomials, solvability by radicals, and geometric constructions.

Unit – I

Algebraic extensions of fields: Irreducible polynomials and Eisenstein criterion – Adjunction of roots – Algebraic extensions – Algebraically closed fields. (Page Nos. 281 – 209)

Unit – II

Normal and Separable extensions: Splitting fields – Normal extensions – Multiple roots – Finite fields – Separable extensions. (Page Nos. 300 – 321)

Unit – III

Galois Theory: Automorphism groups and fixed fields – Fundamental theorem of Galois theory – Fundamental theorem of Algebra. (Page Nos. 322 –339)

Unit – IV

Applications of Galois theory to classical problems: Roots of unity and Cyclotomic polynomials – Cyclic extensions – polynomials solvable by radicals – Symmetric functions –Rule and Compass constructions. (Page Nos. 340 – 364)

Text Book:

Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Cambridge University Press, Second Edition.

Reference Books:

1. Topics in Algebra by I.N. Herstein, Wiley India Pvt. Ltd, New York.

2. Elements of Modern Algebra by Gilbert and Gilbert, PWS Publishing Company Boston.

3. Abstract Algebra by Joffrey Bergen, Academic Press, Boston.

4. Basic Abstract Algebra by Robert B. Ash.

MM 202                                      Semester-II           Paper - II: Lebesgue Measure and Integration

Course Objectives:

The course aims to:

1. To recognize the main drawbacks of Riemann integral and studying the notions of algebra of sets, the class of Borel sets

2. To generalize the notion of length of an interval to the Lebesgue measure of a general set.

3. To understand the relation between measurable functions and measurable sets and establishing the existence of non- measurable sets.

4. To analyze the properties of Lp - spaces.

Course Outcomes: 

After completion of this course, students will be able to

1. Characterize the subsets of R which are Borel sets.

2. Understand the outer measure of a set and determine the Lebesgue measurability of a set.

3. Learn the idea of Lebesgue Integral of a function which is more general in nature and simple to evaluate.

4. Study a class of complete normed spaces namely Lp -spaces.

Unit- I

Algebra of sets - Borel sets - Outer measure - Measurable sets and Lebesgue measure - A non - measurable set - Measurable functions - Littlewood's three principles.

Unit- II

The Riemann integral - The Lebesgue integral of a bounded function over a set of finite measure - The integral of a non - negative function - The general Lebesgue integral.

Unit- III

Convergence in measure - Differentiation of monotone functions - Functions of bounded variation.

Unit- IV

 Differentiation of an integral - Absolute continuity The Lp - spaces - The Minkowski and Holder inequalities - Convergence and completeness.

Text Book:

1. Real Analysis (3rd Edition) (Chapters 3, 4, 5) by H. L. Royden, Prentice-Hall India.

References:

1. Lebesgue measure and Integration by G.de Barra.

2. Measure and Integral by Richard L. Wheeden, Antoni Zygmund.

MM/AM 203 Semester-II Paper- III: Complex Analysis 

Course Objectives:

The course aims to:

1. To introduce the fundamental concepts of complex variables, including regions in the complex plane, limits, Continuity, and differentiability. To develop an understanding of analytic and harmonic functions through the Cauchy- Riemann equations and related theorems.

2. To study elementary complex functions such as exponential, logarithmic, Trigonometric, and hyperbolic functions, including their branches, derivatives, and identities.

3. To provide a thorough understanding of contour integration and related theorems in complex  analysis, including the Cauchy-Goursat theorem, Cauchy Integral Formula, Liouville’s Theorem, and the Fundamental Theorem of Algebra.

4. To enable students to analyze and construct power series and Laurent series representations of complex functions, understand their convergence properties, and apply them to solve problems involving analytic functions and singularities.

Course Outcomes:

After successful completion of this course, students will be able to

1. Understand the fundamental concepts of complex functions, including limits, continuity, differentiability, and analyticity. Apply the Cauchy-Riemann equations and analyze mappings and harmonic functions in the complex plane.

2. Demonstrate knowledge of elementary complex functions such as exponential, logarithmic, trigonometric, and hyperbolic functions, including their branches, derivatives, and identities.

3. Evaluate contour integrals and apply major theorems in complex integration such as the Cauchy-Goursat theorem, Cauchy Integral Formula, Liouville’s Theorem, and the Fundamental Theorem of Algebra.

4. Analyze the convergence of complex sequences and series, and construct Taylor and Laurent series. Apply these series for function representation and manipulation in complex analysis.

Unit-I

Regions in the Complex Plane - Analytic Functions: Functions of a Complex Variable - Mappings - Mappings by the Exponential Function - Limits-Theorems on limits - Limits Involving the Point at Infinity - Continuity - Derivatives - Differentiation Formulas-Cauchy - Riemann Equations - Sufficient Conditions for Differentiability - Polar Coordinates - Analytic Functions - Examples - Harmonic Functions - Uniquely Determined Analytic Functions - Reflection Principle.

Unit - II

Elementary Functions: The Exponential Function- The Logarithmic Function - Branches and Derivatives of Logarithms - some Identities involving Logarithms - Complex Exponents - Trigonometric Functions - Hyperbolic Functions- Inverse Trigonometric and Hyperbolic Functions.

Unit - III

Integrals: Derivatives of Functions 𝑤(𝑡) - Definite Integrals of Functions 𝑤(𝑡) - Contours - Contour Integrals - Some Examples- Examples with Branch Cuts - Upper Bounds for Moduli of Contour Integrals - Antiderivatives

- Proof of The Theorem - Cauchy - Goursat Theorem - Proof of the Theorem - Simply Connected Domains - Multiply Connected Domains - Cauchy Integral Formula - An Extension of the Cauchy Integral Formula - Some Consequences of the Extension - Liouville’s Theorem and Fundamental Theorem of Algebra - Maximum Modulus Principle

Unit - IV

Series: Convergence of Sequences - Convergence of Series - Taylor Series - Proof of Taylor’s Theorem - Examples - Laurent Series - Proof of Laurent’s Theorem - Examples - Absolute and Uniform Convergence of Power Series - Uniqueness of Series Representation- Multiplication and Division of Power Series.

Text Book:

Complex Variables and Applications by James Ward Brown, Ruel V. Churchill McGraw- Hill International Edition

Reference Books:

1. Complex Analysis by Lars V. Ahlfors McGraw- Hill International third Edition

2. Complex Variables Introduction and Applications by Mark J. Ablowitz, Athanassion S. Fokas, Cambridge University Press.

MM/AM 204                                      Semester-II                     Paper – IV: Partial Differential Equations

Course Objectives:

The course aims to:

1. To teach students how to solve first-order nonlinear partial differential equations .

2. To introduce methods for solving linear PDEs with constant coefficients and enable students to classify second- order partial differential equations

3. To provide a thorough understanding of Fourier transforms and their properties.

4. To apply various methods, including Fourier transforms and separation of variables to solve standard PDEs like the diffusion, wave, and Laplace equation

Course Outcomes:

 After successful completion of this course, students will be able to

1. use Cauchy’s method of characteristics and Charpit’s method to solve first-order nonlinear equations.

2. solve linear PDEs with constant coefficients and classify any second-order PDE, then convert it into its canonical form.

3. Compute Fourier transforms of elementary functions and apply the convolution theorem.

4. Solve diffusion, wave, and Laplace equations using Fourier transforms and the separation of variables method.

Unit I:

Nonlinear Equations of the First order - Cauchy’s method of Characteristics -Compatible systems of first order equations - Charpit’s method- Special types of first order equations.

Unit II:

Linear Partial Differential Equations with constant coefficients - Equations with variable coefficients coefficients.

Unit III:

Fourier Transforms : Fourier Integral Representations, Fourier Transforms Pairs, Fourier Transform of Elementary Functions, Properties of Fourier Transform, Convolution theorem, Parseval’s Relation, Transform of Dirac Delta Function, Finite Fourier Transforms.

Unit IV:

Solution of diffusion, wave and Laplace equations by using Fourier transforms and Separation of Variables Method - D’Alembert’s solution of wave equation, Dirichlet problem and Neumann problem.

Text Books:

1. Elements of Partial Differential Equations by Ian Sneddon, Mc.Graw-Hill International Editions,1957. (First and Second Units)

2. Introduction to Partial Differential Equations by K. Shankar Rao, PHI, Third Edition,2011.(Third and Fourth Units)

Reference:

Partial Differential Equations by Lawrence C. Evans, American Mathematical Society, second edition, 2010.

MM 205 Semester-II Paper - V: Topology

Course Objectives:

The course aims to:

1. To introduce students to the foundational concepts of topological spaces, including open sets, bases, subbases, and weak topologies, enabling them to understand abstract topological structures.

2. To develop a deep understanding of compactness and related theorems (such as Tychonoff’s and Ascoli’s) and their applications in analysis and topology.

3. To equip students with knowledge of separation axioms, including T₁, Hausdorff, regular, and normal spaces, and explore their significance through classic theorems like Urysohn’s lemma and Tietze extension theorem.

4. To examine the concept of connectedness in various topological spaces, focusing on components, total disconnectedness, and local connectedness for a broad understanding of continuity and space structure.

Course Outcomes:

After successful completion of this course, students will be able to

1. Explain and illustrate key concepts of topological spaces such as open sets, bases, subbases, and weak topologies using appropriate mathematical examples.

2. Apply compactness and related theorems (Tychonoff’s, Ascoli’s) to analyze topological and metric spaces.

3. Analyze different types of separation axioms and evaluate their implications using theorems like Urysohn’s lemma and Tietze extension theorem.

4. Construct examples and prove results related to connectedness and disconnectedness in topological spaces.

Unit-I

Topological Spaces: The Definition and some examples - Elementary concepts - Open bases and open subbases- Weak topologies. (Page No. 91-106)

Unit-II

Compactness: Compact spaces - Products of spaces - Tychonoff’s theorem and locally compact spaces - Compactness for metric spaces - Ascoli’s theorem. (Page No. 110-128)

Unit-III

Separation: T1 - spaces and Hausdorff spaces - Completely regular spaces and normal spaces - Urysohn’s lemma and the Tietze extension theorem - The Urysohn imbedding theorem.(Page No. 129-141)

Unit-IV

Connectedness: Connected spaces - The components of a space - Totally disconnected spaces - Locally connected spaces. (Page No. 142-152)

Text Book:

Introduction to Topology and Modern Analysis by G.F. Simmon’s Tata Mc Graw Hill Edition.

References:

Introductory Topology by Mohammed H. Mortad. Explorations in Topology by David Gay.

    Topology by Munkers’

    Elementary Topology by Michael C. Gemignani.