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.
• Bézout’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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