Preliminaries

Sets and cardinality
Equivalence relations

Groups

Definitions and examples, permutation groups
Subgroups, group homomorphisms, kernels
Cayley’s theorem, Lagrange’s theorem
Normal subgroups, quotient groups
Isomorphism theorems
Group actions, the class equation and applications
Sylow Theorems
Semidirect Products
The category of groups, products, coproducts

Rings

Definitions and basic examples, subrings, homomorphisms
Ideals, maximal and prime ideals
Polynomial rings
Quotient rings
Integral domains
Localization, field of fractions
Principal ideal domains, unique factorization domains, euclidean domains
Chinese remainder theorem
Spectrum of a ring
Modules over a ring
Classification of modules over a PID

Linear Algebra

Linear transformations, matrices
The determinant
Eigenvalues, eigenvectors, diagonalization
Proof of Jordan canonical form
Tensor products
Inner product spaces
Hermitian forms
Adjoint operators
Spectral theorem for normal operators

Fields

Field extensions
Algebraic closure
Galois Theory
Solvability
Finite fields