## Introduction to Graduate Algebra

The notes, videos and problems on this page are intended as a way to remind basic constructions and results in algebra. If you have never seen a particular topic, it does make sense to read more about it in a textbook, such as Dummit and Foote's "Algebra".

APOLOGY: the videos below have poor audio/video synchronization.

- Groups and homomorphisms: video and notes (G-1).
- Subgroups, cosets and normal subgroups: video, notes included in G-1 above.
- First Isomorphism Theorem and Correspondence Pinciple: video, notes included in G-1 above.
- Lagrange's Theorem and examples: video, notes included in G-1 above.
- Actions and orbits, orbit-stabilizer formula: video and notes (G-2).
- Euclidean Division. Cyclic groups, their subgroups and quotients: video, notes included in G-2 above.
- Examples of groups: Dn, Sn, An, GLn(F), SL(F), Q8: video, notes included in G-2 above.
- Direct sums, free abelian groups, classification of f.g. abelian groups (without proof): video, notes included in G-2 above.
- Cauchy's and Sylow's Theorems: video and notes (G-3).
- Properties of the symmetric group: video, notes included in G-3 above.
- Classification of groups of small order: video., notes included in G-3 above.
- Rings, homomorphisms, ideals and quotients: video and notes (R-1)
- Factorization in commutative rings: primes and irreducibles: video, notes included in R-1 above.
- Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains: video, notes included in R-1 above.
- Prime and maximal ideals. Fields of fractions: video and notes (R-2).
- Polynomial rings and irreducibility criteria. Primes in Z[i] : video, notes included in R-2 above.
- Gauss Lemma. R[x] is a UFD if R is. General CRT: video, notes included in R-2 above.
- Modules and their homomorphisms. Quotient Modules and Finitely generated modules: video and notes (R-3).
- Free modules and ranks. Matrices, change of basis, determinants and traces, Cramer's Rule: video, notes included in R-3 above.
- Submodules in free modules over PID. Smith Normal Form of a matrix over PID: video, notes included in R-3 above.
- Modules over PIDs: classification theorem. Elementary divisors and invariant factors. Examples: video, notes included in R-3 above.
- Vector spaces, bases, dimensions, rank-nullity formula: video and notes (LA-1)
- Eigenvalues and eigenvectors. Diagonalizability: video, notes included in LA-1 above.
- Rational canonical form and Jordan canonical form: video, notes included in LA-1 above.
- Vector spaces with a scalar product: Euclidean and Hermitian case, orthogonality and Gram-Schmidt orthogonalization process: video and notes (LA-2)
- Duals and double duals, Riesz Representation Theorem, adjoint operators: video, notes included in LA-2 above.
- Spectral Theorem for normal operators (finite dimension, complex and real cases). Special cases: self-adjoint, skew-adjoint, unitary and orthogonal operators: video, notes included in LA-2 above.
- Field extensions, degree of extension, multiplicative property of degrees, algebraic and transcendental elements: video and notes (FE-1).
- Separable polynomials, splitting fields, algebraic closures. Finite multiplicative subgroup of a field is cyclic: video, notes included in FE-1 above.
- Finite fields: existence and uniqueness: video, notes included in FE-1 above.

Problem lists (borrowed from course Math 206) for discussion and self-study: **groups**, **rings**, linear algebra, fields

Homework problems: **groups**, **rings**, **linear algebra and fields**

Online **notes** from MIT on groups and rings.