Stellar

Sets Naive set theory Definitions Basic results Relations Types of orders Examples Functions Basic results Set Cardinality Integers Principle of induction Exercises Construction of the integers Division Greatest common divisor Prime numbers Diophantine equations Exercises Modular Arithmetic Divisibility rules Exercises Euler's totient function Exercises Chinese remainder theorem Exercises Miscellaneous Exercises Binary operations Examples Monoids Introduction to monoids Submonoids Groups Introduction to groups Symmetric group Subgroups Exercises Generating set of a group Types of subgroups Geometric groups Cyclic groups Examples Permutations and k-cycles Exercises Cosets Isomorphisms Kernels Homomorphisms from cyclic groups Exercises Group actions Conjugacy in permutation groups Quotient groups Isomorphism theorems Exercises Direct and semidirect product Direct product Semidirect product Sylow preliminaries Sylow theorems Applications Exercises Fields Introduction to field Rings Introduction to rings Exercises Subrings and ideals Exercises Algebras over fields Exercises Ring homomorphisms Exercises Quotient rings Isomorphism theorems Fields and fraction fields Ideals in commutative rings Comaximal ideals and Chinese remainder theorem Noetherian rings Euclidean domains Exercises Unique factorization domains Failure of unique factorization in ℤ[√−5]Ideal factorization in ℤ[√−5]Quadratic integer rings Polynomials Polynomials over rings Valuations associated with prime elements Primitive polynomials Exercises Field extensions Exercises Modules Modules over a ring Submodules, morphisms and quotients Generators Exercises Torsion modules and cyclic modules Modules over PIDs Primary modules Socle and cyclic p-primary components Exercises Free modules over PIDs Classification of finitely generated modules over PIDs Classification of finite abelian groups Elements of prescribed order Exercises Endomorphisms and K[X]-modules Cyclic subspaces and companion matrices Primary and rational canonical forms Jordan canonical form Powers and ranks of Jordan blocks Exercises Minimal polynomial Minimal polynomial and canonical forms Polynomials of matrices Exercises Lattices References

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