Algebra 1 06-DALGLM1

After completing the course a student should know the following topics and have proficiency in their applications:
Basic algebraic structures (groups, rings, fields).
Basic notions of the group theory (subgroups, cosets, Lagrange's theorem).
Factor groups (construction, the First Isomorphism Theorem).
Direct sums and products of groups.
Cyclic groups (classification, subgroups, homomorphic images).
Symmetric groups (Cayley's theorem).
Basic notions of the ring theory (subrings, zero divisors, units, homomorphisms).
Ideals and factor rings.
Commutative rings (prime and maximal ideals, Chinese Remainder Theorem).
Rings of quotients and localizations.
Polynomial rings (roots, Bezout's theorem).
Elements of divisibility theory (uniform factorization, principal ideal domains, Euclidean rings).
Field extensions (basis and degree, algebraic extensions).