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Indian Institute of Science Education and Research Bhopal

Mathematics

MTH 401: Fields and Galois Theory (4)

Pre-requisites: MTH 301 Groups and Rings

Learning Objectives:

Field Extensions are studied in an attempt to find a formula for the roots of polynomial equations, similar to the one that exists for a quadratic equation. The Galois group is introduced as a way to capture the symmetry between these roots; and the solvability of the Galois group determines if such a formula exists or not. In the 19th century, Galois proved that a formula does not exist for a general 5th degree equation. More importantly, the use of groups to study the symmetry of other objects is a pervasive theme in Mathematics, and this is traditionally the first place where one encounters it.
The topics to be covered include irreducibility of polynomials, Field Extensions, Normal and Separable Extensions, Solvable Groups, and Solvability of polynomial equations by radicals, Finite fields, and Cyclotomic fields

Course Contents:

Polynomial rings, Gauss lemma, Irreducibility criteria

Definition of a field and basic examples, Field extensions

Algebraic extensions and algebraic closures

Classical Straight hedge and compass constructions (optional)

Splitting fields, Separable and Inseparable extensions

Cyclotomic polynomials, Galois extensions

Fundamental theorem of Galois theory

Composite and Simple extensions, Abelian extensions over Q

Galois groups of polynomials, Solvability of groups, Solvability of polynomials

Computations of Galois groups over Q

Suggested Books:

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