Algebra Notes for BS/MSc
Algebra is a fundamental branch of mathematics that deals with symbols, variables, and the rules for manipulating these symbols to solve equations and study mathematical structures. It plays a crucial role in various fields of mathematics, science, engineering, and everyday life.
Algebra is a broad area of mathematics that includes various branches such as abstract algebra, linear algebra, and algebraic geometry.
Abstract Algebra:
- Groups:
- Definition and basic properties of groups.
- Subgroups, cosets, and Lagrange’s theorem.
- Cyclic groups, permutation groups.
- Normal subgroups and quotient groups.
- Isomorphism theorems.
- Rings:
- Definition and basic properties of rings.
- Subrings and ideals.
- Quotient rings and isomorphism theorems.
- Polynomial rings.
- Fields:
- Definition and basic properties of fields.
- Finite fields.
- Algebraic extensions and algebraic closure.
- Field automorphisms and Galois theory.
- Modules:
- Definitions and basic properties of modules.
- Submodules, quotient modules.
- Free modules and bases.
- Tensor products of modules.
- Linear Algebra:
- Vector spaces, subspaces.
- Linear independence and basis.
- Dual spaces and dual bases.
- Linear transformations and matrices.
- Eigenvalues and eigenvectors.
Algebraic Geometry:
- Affine and Projective Varieties:
- Definition and basic properties.
- Ideals and algebraic sets.
- Affine and projective spaces.
- Coordinate Rings and Morphisms:
- Coordinate rings of algebraic sets.
- Morphisms between varieties.
- Rational maps and birational equivalence.
- Schemes:
- Definition and basic properties of schemes.
- Sheaves and their cohomology.
- Structure sheaf and locally ringed spaces.
- Intersection Theory:
- Intersection multiplicities.
- Bézout’s theorem.
Additional Topics:
- Homological Algebra:
- Homomorphisms, kernels, cokernels.
- Chain complexes and homology.
- Ext and Tor.
- Representation Theory:
- Group representations.
- Characters and character tables.
- Maschke’s theorem.
- Lie Algebras:
- Definition and basic properties.
- Representations of Lie algebras.
- Simple and semisimple Lie algebras.
- Category Theory and Algebra:
- Basic concepts of category theory.
- Functors, natural transformations.
- Limits and colimits.
These topics provide a broad overview of algebra at the advanced undergraduate and graduate levels. The specific content covered may vary between institutions and courses. Algebra plays a fundamental role in various branches of mathematics and has wide-ranging applications in other scientific disciplines.
Algebra Notes for BS/MSc
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