Abstract Algebra Exam 3 Review Problems and Solutions (Basic Ring Theory and Field Theory)
Types of Abstract Algebra Practice Questions and Answers: 1) Classify finite Abelian groups, 2) Definitions of ring, unit in a ring, zero divisor, field, ideal in a ring, principal ideal, principal ideal domain, prime ideal, maximal ideal, 3) ring homomorphisms, 4) true/false questions about IDs, fields, PIDs, UFDs, and EDs, 5) irreducibility and reducibility of polynomials over fields, 6) proofs about ideals, 7) an integral domain that is not a unique factorization domain. https://amzn.to/2ZqLc1J ("Contemporary Abstract Algebra", by Joe Gallian) 🔴 Abstract Algebra Course Playlist: • Abstract (Modern) Algebra Course Lectures 🔴 Real Analysis Course Playlist: • Introduction to Real Analysis Course Lectures 🔴 Complex Analysis Course Playlist: • Introduction to Complex Analysis Course Le... #abstractalgebra #abstractalgebrareview #ringtheory Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter: / billkinneymath 🔴 Follow me on Instagram: / billkinneymath 🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ (0:00) Types of problems (0:35) Abelian groups of order 72 (isomorphism classes) (3:20) Number of Abelian groups of order 2592 (use partitions of integer powers) (7:02) Definition of a ring R (11:33) Definition of a unit in a commutative ring with identity (12:54) Definition of a zero divisor in a commutative ring (14:56) Definition of a field F (could also define an integral domain) (17:04) Definition of an ideal of a ring (two-sided ideal) (19:04) Ideal Test (20:28) Principal Ideal definition (22:14) Principal Ideal Domain (PID) definition (23:36) Prime Ideals, Maximal Ideals, and Factor Rings (Quotient Rings). Relationship to integral domains and fields. (29:29) Irreducible element definition (in an integral domain) (31:40) Z8 units and zero divisors, U(Z8) group of units (34:09) Ring homomorphisms from Z12 to Z20 (38:38) Integral domains, fields, PIDs, UFDs, EDs (True/False) (41:15) Z[x] is a UFD but not a PID (Z[x] is a Unique Factorization Domain but not a Principal Ideal Domain) (42:29) Long division in Z3[x] (& synthetic division mod 3) (Division algorithm over a field) (48:36) Reducibility test of degree 2 polynomial over field Z5 (50:43) Eisenstein's Criterion for irreducibility over the rationals Q (52:53) Tricky factorization to prove reducibility over Q (54:19) Mod p Irreducibility test for degree 3 polynomial over Q (57:11) Prove fields have no nontrivial proper ideals (1:01:21) Prove the intersection of ideals is an ideal (use the Ideal Test) (1:05:32) Mod p Irreducibility test for degree 4 polynomial over Q (1:12:33) Factor ring calculations in Z3[x]/A, where A is a maximal principal ideal generated by an irreducible polynomial over Z3 (1:25:12) Part of proof that Z[sqrt(-5)] is not a UFD (it's an Integral Domain that is not a Unique Factorization Domain). Need properties of a norm defined on Z[(-5)^(1/2)] and the definition of irreducible in an integral domain. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.
