# Algebra Syllabus (for MS and Preliminary Exams)

#### Group Theory

• Subgroups, Lagrange's theorem
• Normal subgroups, quotient groups
• Isomorphism theorems, permutation groups, simplicity of An
• Cyclic groups, direct products (sums)
• Finitely-generated Abelian groups, p-groups, Sylow theorems

#### Vector Spaces and Modules

• Submodules, quotient modules, isomorphism theorems
• Linear independence, bases, linear operators, homomorphisms
• Rank, determinant, finitely-generated modules over PID's
• Inner product spaces, orthogonality (Gram-Schmidt)
• Dual spaces, determinants, characteristic & minimal polynomials
• Eigenvalues and eigenvectors, Cayley-Hamilton theorem
• Canonical forms (triangular, rational, Jordan)

#### Rings

• Subrings, ideals, quotient rings, isomorphism theorems
• Arithmetic of Z and Zn (Fermat's theorem, Chinese Remainder theorem)
• Integral domains and quotient fields
• Prime and maximal ideals, euclidean rings, PID's and UFD's
• Polynomial rings, Gauss' lemma

#### Fields

• Finite and algebraic extentions, Galois extensions
• Simple extensions, finite fields
• Galois theory (in characteristic 0), geometric constructions