Cogalois Theory

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Understanding of the relation between symmetries of roots of a polynomial and its solubility in terms of simple algebraic formulae; working knowledge of interesting group actions in a nontrivial context; working knowledge, with applications, of a nontrivial notion of finite group theory soluble groups ; understanding of the relation between algebraic properties of field extensions and geometric problems such as doubling the cube and squaring the circle. Review of polynomial rings, factorisation, integral domains.

Reminder that any nonzero homomorphism of fields is injective. Fields of fractions. Review of group actions on sets, Gauss' Lemma and Eisenstein's criterion for irreducibility of polynomials, field extensions, degrees, the tower law. Symmetric polynomials. Separable extensions. Splitting fields and normal extensions. The theorem of the primitive element.

NPTEL :: Mathematics - NOC:Galois Theory

The existence and uniqueness of algebraic closure proofs not examinable. Examples: Kummer extensions, cyclotomic extensions, finite fields and the Frobenius automorphism. Techniques for calculating Galois groups. Soluble groups. Solubility by radicals, solubility of polynomials of degree at most 4, insolubility of the general quintic, impossibility of some ruler and compass constructions.

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Galois theory

You are here Home. Lecturer s :. Algebraic Number Theory - Kyoto Algorithmic Number Theory Algorithms for Diophantine Equations. Analytic Number Theory - Goldstein. Analytic Number Theory-Jia, Matsumoto.

Analytic Number Theory - Proceedings of a conference in hono. Analytic Number Theory - Rademacher. An Introduction to Algebraic Number Theory. Applications of Automorphic Forms to Number Theory. Arithmetic and Geometry I. Arithmetic and Geometry II. Arithmetic Fundamental Groups. Arithmetic Theory of Elliptic Curves.


Automorphic Forms and Number Theory - Shimura. Basic Analysis of Regularized Series and Products.


Basic Structures of Function Field Arithmetic. Calculus 1 Tom A Apostol. Calculus 2 Tom A Apostol. Class Field Theory - it Centenary and Prospects. Classical Diophantine Equations. Cohomologie, Stabilisation et Changement de Base.

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Collected papers of G. Hardy V. Computational Methods in Number Theory.