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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