Algebra Notes for BS/MSc

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:

  1. Groups:
  • Definition and basic properties of groups.
  • Subgroups, cosets, and Lagrange’s theorem.
  • Cyclic groups, permutation groups.
  • Normal subgroups and quotient groups.
  • Isomorphism theorems.
  1. Rings:
  • Definition and basic properties of rings.
  • Subrings and ideals.
  • Quotient rings and isomorphism theorems.
  • Polynomial rings.
  1. Fields:
  • Definition and basic properties of fields.
  • Finite fields.
  • Algebraic extensions and algebraic closure.
  • Field automorphisms and Galois theory.
  1. Modules:
  • Definitions and basic properties of modules.
  • Submodules, quotient modules.
  • Free modules and bases.
  • Tensor products of modules.
  1. Linear Algebra:
  • Vector spaces, subspaces.
  • Linear independence and basis.
  • Dual spaces and dual bases.
  • Linear transformations and matrices.
  • Eigenvalues and eigenvectors.

Algebraic Geometry:

  1. Affine and Projective Varieties:
  • Definition and basic properties.
  • Ideals and algebraic sets.
  • Affine and projective spaces.
  1. Coordinate Rings and Morphisms:
  • Coordinate rings of algebraic sets.
  • Morphisms between varieties.
  • Rational maps and birational equivalence.
  1. Schemes:
  • Definition and basic properties of schemes.
  • Sheaves and their cohomology.
  • Structure sheaf and locally ringed spaces.
  1. Intersection Theory:
  • Intersection multiplicities.
  • B├ęzout’s theorem.

Additional Topics:

  1. Homological Algebra:
  • Homomorphisms, kernels, cokernels.
  • Chain complexes and homology.
  • Ext and Tor.
  1. Representation Theory:
  • Group representations.
  • Characters and character tables.
  • Maschke’s theorem.
  1. Lie Algebras:
  • Definition and basic properties.
  • Representations of Lie algebras.
  • Simple and semisimple Lie algebras.
  1. 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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