algebra (hs26)

algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems.

1. review of linalg i & ii (set theory, composition laws)
2. group theory (subgroups, order, quotient by subgroup, normal subgroups, cyclic groups, simple groups, group automorphisms)
3. group actions and sylow thms (actions, orbits, stabilizers, sylow's thm)
4. symmetric groups (definition & properties, permutations, signatures of permutations, conjugacy classes, simplicity of \(A_n\))
5. rings and modules (definition & properties, examples, polynomial rings)
6. commutative algebra (prime & maximal ideals, noetherian rings, principal ideal domains, prime decomposition in PIDs, chinese remainder thm, gcd & lcm, classification of modules over PIDs)