Course Type | Course Code | No. Of Credits |
---|---|---|
Foundation Elective | SUS1MA539 | 4 |
Semester and Year Offered: Winter semester
Course Coordinator and Team: Geetha Venkataraman and others
Email of course coordinator: geetha[at]aud[dot]ac[dot]in
Pre-requisites: Mathematics at the XII grade level and 4 credits of Algebra III (SUS1MA509)
Aim: This is a fourth course in Algebra and is intended for those who wish to deepen their base in the subject. This course will help students to enhance their foundational abilities developed through the three compulsory Algebra courses. Such skills would also add to students’ ability to analyse and solve problems even in real life. This course is an optional course primarily intended for students interested in pursuing post-graduation in Mathematics.
Course Outcomes:
After completing this course, depending on which 2 modules are taken (see syllabus below), students will be able to
Brief description of modules/ Main modules:
There are three modules in this course, namely
Module 1: Group Theory
Module 2: Field Theory
Module 3: Linear Algebra
A student has to opt for any two modules.
Module 1: Group Theory
Definition and examples of group actions, Stabilizers and kernels of group actions, Permutation representation associated with a given group action. Applications of group actions, Class equation and consequences, conjugacy in Sn, p-groups, Sylow’s theorems and consequences, Definition and examples of simple groups, non-simplicity tests.
Module 2: Field Theory
Extension Fields: Definition of Field Extension, Fundamental Theorem of Field Theory (Kronecker’s Theorem), Splitting Fields: Definition, Existence, uniqueness and examples, Zeros of a polynomial: Criterion for multiple zeros of a polynomial, zeros of an irreducible polynomial, perfect fields and perfect fields and zeros of an irreducible polynomial.
Algebraic Extensions: Types of Extensions and Characterization of Extensions and minimal polynomial, Degree of Extensions: Relationship between finite and algebraic extensions, Tower Theorem, Primitive Element Theorem and Properties of Algebraic Extensions.
Finite Fields: Classification of Finite Fields, Structure of Finite Fields and examples and Subfields of a Finite Field.
Geometric Constructions: Constructible numbers: Definition, constructible numbers form a subfield of the real numbers, If c is a constructible number the extension of the rational numbers by c over the Rational numbers has degree a power of 2, Examples and classical problems trisecting and angle, squaring a circle and finding a cube of double the volume of a given cube.
Module 3: Linear Algebra
Linear functionals, Double dual, Transpose of a Linear transformation, Characteristic Values and Diagonalization, Annihilating Polynomials, minimal polynomials and Invariant subspaces, Direct sum decompositions and Spectral Theorem, Primary decomposition.
Inner Products and Inner Product spaces, Linear functionals and Adjoint, Unitary operators and Normal operators.
Assessment Details with weights:
S. No. | Assessment | Date/period in which Assessment will take place | Weightage |
1 | Class test | First week of February | 10% |
2 | Mid Semester Exam | Early March | 25% |
3 | Home Assignments/ Tutorials | Throughout the semester | 15% |
4 | Presentation/ Viva | Early April | 15% |
5 | End Semester Exam | End April-early May | 35% |
Reading List:
ADDITIONAL REFERENCE: