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Course Contents

Fields: definition and examples. Ring of polynomials over a field. Field extensions. Algebraic and transcendental elements, Algebraic extensions. Splitting field of a polynomial. Algebraic closure of a field, Uniqueness. Normal, separable, purely inseparable extensions. Primitive elements of a field extension. Simple extensions. Fundamental theorem of Galois. Solvability by radicals Solutions of cubic and quartic polynomials, In solvabity of quintic and higher degree polynomials. Geometric constructions. Cyclotomic extensions. Finite fields.Cyclotomic polynomials and its properties. Traces and norms. Modules definition, examples and basic properties. Free modules, submodules andquotient modules, isomorphism theorems. Localization. Direct sum and directproducts. Noetherian and Artinian rings and modules, structure of Artinian rings,Hilbert basis theorem. Jordan Holder theorem. Radicals of modules, Nakayamalemma.