Comps Syllabus for Algebra
Download algebra comps syllabus
Sets, Functions, and Integers
- One-to-one, onto and bijective maps
- Equivalence relations and equivalence classes
- Division algorithm, gcd and lcm, primes and prime factorization
Groups
- Uniqueness of identities and inverses
- The order of an element
Subgroups
- Lagrange's Theorem and its consequences
- Cosets
- Normal subgroups
- Quotient groups
Group Homorphisms
- Kernels and images
- Isomorphisms
- The Fundamental Theorem of Group Homomorphisms: G/Ker(φ)≅Im(φ)
Permutations
- Sn and disjoint cycle decomposition
- Transpositions and An
Rings
- Commutative rings; Rings with unity; Fields
- Polynomial rings
Ideals
- Ideals
- Quotient rings
Ring Homomorphisms
- Kernels and images
- Isomorphisms
- The Fundamental Theorem of Ring Homomorphisms: R/Ker(φ)≅Im(φ)
Quotient Rings and Fields
- Criteria for R to be a field
- Maximal ideals
- Criteria for R/M to be a field
Polynomial Rings k[x], for a Field k
- The division algorithm
- Every ideal in k[x] is principal
- Irreducible polynomials and maximal ideals in k[x]