Advanced Algebra

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

1. identify group actions, stabilizers and orbits, simple groups, use/apply their properties;
2. prove significant results in group theory using group actions, like Cauchy’s Theorem, Sylow’s Theorem, class equation etc.
3. identify field extensions, splitting fields of polynomials, algebraic extensions, finite fields and use/apply their properties;
4. prove and apply the fundamental theorem of field extensions;
5. decide which numbers are constructible and use/ apply the concept;
6. identify linear functionals, their dual, transpose of linear transformations, inner product spaces, adjoints, normal and unitary operators, use and apply their properties;
7. calculate characteristic value, decide diagonalizability, work out annihilating and minimal polynomials; use and apply their properties;
8. prove the primary decomposition and spectral theorem and apply them.

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:

Reading List:

• David S. Dummit and Richard M. Foote, Abstract Algebra, Third Edition (2004), John Wiley and Sons.
• Joseph A. Gallian, Contemporary Abstract Algebra, (Fourth Edition), Narosa Publishing House, New Delhi, 1999.
• Jimmie Gilbert and Linda Gilbert, Linear Algebra and Matrix Theory, (Second Edition), Brooks Cole, 2004.

ADDITIONAL REFERENCE:

• Bhattacharya, Jain and Nagpal, Basic Abstract Algebra (Second Edition), Cambridge, 2009.
• Kenneth Hoffman and Ray Kunz, Linear Algebra, (Second edition), PHI, 2009.
• Friedberg H. Stephen, Insel J. Arnold, Spence E. Lawrence, Linear Algebra, (Fourth Edition), PHI, 2002.