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Higher Algebra I Course Notes by Eyal Z. Goren pdf

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Higher Algebra I Course Notes written by Eyal Z. Goren, Mcgill University This is an other great mathematics book cover the following topics.

Groups
Our starting point, Group actions, The class equation, p-groups, Examples of p groups, Groups of order p, Groups of order p2, Groups of order p3, The coset representation, The Sylow theorems, Examples and applications, Semi-direct product, Application to groups of order pq., Cases where two semi-direct products are isomorphic, Groups of small order, The Cauchy-Frobenius formula and applications to combinatorics, Applications to combinatorics, Simplicity of PSLn(Fq )

The Jordan-H¨older theorem and solvable groups
Composition series and composition factors, Jordan-H¨older Theorem, Solvable groups

Free groups and free products
Categories: Definition of a category, Example: Sets, Example: Gp, Example: AbGp, Example: VSpk, Example: 2S, Categories: initial and final objects, The opposite category, Free groups, Categories: universal objects, Free groups: further properties, Generators and relations, Free products, Category theory: functors and adjoint functors, Examples of functors, Adjoint functors

Modules
Recall, Localization of rings and modules, On localization of rings, The field of fractions, On localization of modules, Ideals under localization, Free modules and rank, Local properties, Equivalence of categories, Definition of a natural transformation, Definition of equivalence, Some examples, A criterion for equivalence, Modules over a PID, Applications of the Structure theorem for modules over PID, Finitely generated abelian groups, Vector spaces, The Jordan canonical form, Morita equivalence, Injective and projective limits, More Examples: injective limits, More Examples: projective limits

Infinite Galois theory
A quick review of Galois theorey of finite extensions, First definitions, The main theorem of Galois theory, Zp, Hensel’s lemma, Finite fields, Cyclotomic fields

Kummer Theory
Cyclic Galois extensions, Kummer extensions, Characters of finite groups, Kummer extensions

Calculation of Galois groups
The discriminant, Calculating Galois groups by reduction modulo

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