Algebra II * 1000-134AG2*
1. Categories and functors, isomorphism. Examples: Vect, Top, Gr, Ab, one object categories, sheaves, etc. Product and direct sum in a category, product and direct sum in categories
groups and abelian groups. (1-2 lectures)
2. Commutator of elements, commutant of a group, abelianization, Solvable groups, simple groups, solvability of S_n, n < 5, simplicity of A_n, n > 4., semisimple product, exact sequences, splitting, examples (2-3 lectures).
3. Field extensions, the group of automorphism of field extension., algebraic extensions, extensions by a root of a polynomial, splitting field of a polynomial, normal extensions, universal property of normal extensions. The algebraic closure of a field - construction and the uniqueness, Examples. (2-3 lectures).
4. Roots of unity. Existence and uniqueness of fields of p^n elements.
5. Galois Theory of finite extensions in characteristic 0. Irreducible polynomials in characterictic 0 have no multiple roots. Theorem of Abel, Automorphisms of extensions, Galois extensions. Main Theorem of Galois Theory (2-3 lectures).
6. Applications of Galois theory: the Fundamental Theorem of Algebra, solvable extensions, solving equations by radicals (1-2 lectures)
7. Applications of Galois theory: constructability in geometry
8. Modules, torsion elements, direct sum, finitely generated modules, free modules. Homomorphisms of modules, the kermel, factor module, exact sequences of modules, splittings. Classification of finitely generated modules PID. Corollaries :: classification of finitely generated abelian groups,
Jordan's Theorem from linear algebra on canonical form of matrices (2-3 lectures).
9. Tensor product of modules, exterior power of a module, exterior algebra.
Remark. Topics 4, 7 and 9 are optative. For example, they can be discussed in class only.
Type of course
Bibliography
S. Lang, Algebra
L. Rowen, Algebra: groups, rings and fields, Wellesley, Massachusetts 1994
Additional information
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