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" or M. Artin's "Algebra" or E.B. Vinberg's "A course in Algebra" or S. Lang's "Algebra". Also see links at the bottom of this page.
APOLOGY: some of the videos below have poor audio/video synchronization.
First Isomorphism Theorem and Correspondence Principle: video, notes.
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.
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.
If you have are a registered UC Irvine student and go online through UCI WebVPN, you should be able to download Springer textbooks for free. Follow this link first https://guides.lib.uci.edu/eBooks/ebooks_science and then on to "Springer Online Books". Then you can search, e.g. for Gorodentsev "Algebra I", or Lang 'Undergraduate Algebra".