Groups
Basic Definitions and Examples
• Group; Abelian group; order of a group; order of an element of a group
• The dihedral group D_2n of order 2n; the symmetric group S_n; the matrix groups GL_n(F)
Subgroups, Homomorphisms and Quotient Groups
• Group homomorphisms and isomorphisms; kernels and images; group actions
• Subgroups, normal subgroups, lattice of subgroups
• Quotient groups; left (or right) cosets; Lagrange's Theorem; Cauchy's Theorem; The Isomorphism Theorems
• Fundamental Theorem of Finitely Generated Abelian Groups
• Free groups, groups with relations
• Sylow Theorems

Rings
Basic Definitions and Examples
• Rings; division rings; fields; zero divisors; units; integral domains; subrings; group of units; Z, Q, R, and C; Z_n (integers modulo n); ; matrix rings; R[x]
Ring Homomorphisms, Ideals and Quotients
• Ring homomorphisms, kernels and images; ideals; quotient rings; The Isomorphism Theorems; lattice of ideals; principal ideals; maximal ideals; prime ideals
Polynomial Rings, Factorization and Irreducibility
• Polynomial rings R[x], R[x₁,x₂, ... , x_n]; the division algorithm and the Euclidean algorithm; uniqueness of factorization; principal ideals in F[x]; irreducibility; Gauss' Lemma; Eisenstein's Criterion; maximal ideals in F[x]

Fields
Field Extensions
• Characteristic; prime subfield; extension field; degree of an extension; basis for F[x]/(p(x)); simple extension; algebraic extensions; minimal polynomial; splitting fields, finite fields, impossibility of certain geometric constructors.
• Galois Groups of quadratic and cubic polynomials

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