17fall_519

Algebraic Structures I

- Fall 2017 -

Instructor: Kwangho Choiy

Course Website: https://sites.google.com/site/kchoiy/home/teaching/f17_519

Class Meeting: MWF 12:00pm - 12:50pm in Neckers 218.

Textbook: Abstract Algebra 3rd edition by David S. Dummit and Richard M. Foote. 

Course Objectives/Topics: Our goal is to learn the theory of groups and rings. The topics include: structure of groups, subgroups, quotient groups, isomorphism theorems, group actions, Sylow theorems, semi-direct product of groups, classification of finite groups, structure of rings, ideals, UFD, PID, Euclidean domain, polynomial rings, irreducibility criteria.

Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule linked [here], updated on 8/27/2017.

Exams: There will be two mid-term exams and one final in the classroom.

Each HOMEWORK ASSIGNMENT will be posted as below at least one week ahead of the due date (see the Course Schedule linked above for a tentative assignment schedule!):

*HW Policies: You should show all your work and submit it in class on the due date. No late homework will be accepted.*

Updates and Remarks - MATH519 - Fall 2017:

[Dec 8, 2017] Lecture 43:  Chapter 9 - sketch proof of R[x]/I[x] isom to (R/I)[x], example, Eisenstein Criterion for Z and for ID and application, ideas for the proof of UFD[x] being UFD, its applications, Gauss Lemma, and further arguments related to factorizations in R[x] and Quot(R)[x].

   -- Graded HW 9 and Special Assignment are returned.

[Dec 6, 2017] Lecture 42:  Chapter 9 - definition of polynomial rings, properties, examples, UFD[x] is UFD, F[x] is ED with sketch of proof.

   -- Solution of HW 9 is handed out.

[Dec 4, 2017] Lecture 41:  Chapter 8 - more discussions related to prime/irreducible elements; UFD,PID,ED, more examples.

   -- A PDF file of practice problems, selected topics, and details for FINAL is E-mailed.

[Dec 1, 2017] Lecture 40:  Chapter 8 - Z[sqrt{-5}] is not UFD, Examples in factorization in Z[i], more arguments related to prime/irreducible elements, Discussed questions in HW 9.

[Nov 29, 2017] Lecture 39:  Chapter 8 - proved PID is UFD, introduced ACC, Some arguments related to prime/irreducible elements, gave partial solution for HW 9 questions.

[Nov 27, 2017] Lecture 38:  Chapter 8 - prime element, irreducible element, examples, outlined their properties/relations.

   -- Graded HW 8 with solution is returned

[Nov 20, 2017] Lecture 37:  Chapter 8 - proved PID is UFD, an example for UFD, not PID with proof, detailed about the notion of uniqueness.

   -- HW 9 (due on Dec. 1, Fri) is handed out.

   -- Special Assignment (due on Dec. 4, Mon; voluntary to submit) is E-mailed.

[Nov 17, 2017] Lecture 36:  Chapter 8 - further concrete examples for ED, relationship between 3 properties and ED,PID,UFD, defined UFD, example for PID, proof of Z being PID.

[Nov 15, 2017] Lecture 35:  Chapter 8 - discussed ED => PID => UFD but not the other directions, concrete examples for ED.

[Nov 13, 2017] Lecture 34:  Chapter 8 - 3 main properties integers satisfy, introduced how to expand such properties to abstract ring-related notions, introduced ED, PID, UFD, defined ED, PID, examples.

   -- Graded HW 7 with solution is returned

[Nov 11, 2017] An additional note for Chinese Remainder Theorem is E-mailed.

[Nov 10, 2017] Lecture 33:  Chapter 7 - motivating question for Chinese Remainder Thm, comaximal, Proof of Chinese Remainder Thm, remarks, examples.

   -- HW 8 (due on Nov. 20, Mon) is handed out.

[Nov 8, 2017] Lecture 32:  Chapter 7 - some properties of ideals, principal/prime/maximal ideals, their properties related to ID, field, examples.

[Nov 6, 2017] Lecture 31:  Chapter 7 - more about kernel, ideals, factor rings, examples, isomorphism theorem.

[Nov 3, 2017] Lecture 30:  Chapter 7 - properties of rings, examples, ring homomorphism and its properties, kernel of the ring homomorphism, ideal. 

   -- HW 7 deadline is now Nov. 10th.

[Nov 1, 2017] Lecture 29:  Chapter 7 - introduced several kinds of rings, examples, diagram. 

   -- Graded Exam 2 is returned // HW 7 (due on Nov. 8, Wed) is handed out.

[Oct 30, 2017] Lecture 28:  Chapter 7 - introduced a ring, definitions, examples, new terms.

[Oct 27, 2017] Lecture 27: Exam 2 taken.

   -- Exam 2 solution is handed-out.

[Oct 25, 2017] Lecture 26:  Chapter 5.2 - fundamental thm for finite ab gp; f.g. abelian gp, remarks, examples, some useful arguments.

   -- A PDF file of Partial Solutions for practice problems for EXAM 2 is E-mailed // An additional note for further topics for group theory is E-mailed. // Graded HW 6 with solution is returned

[Oct 23, 2017] Lecture 25:  Chapter 5.5 - recall the def of semi-direct products, remarks, examples.

   -- A PDF file of practice problems, selected topics, and details for EXAM 2 is E-mailed.

[Oct 20, 2017] Lecture 24:  Chapter 4 - introduced further topics in group theory, three special group actions, conjugacy classes, class equation, examples / Chapter 5.5 - define a semi-direct product.

[Oct 18, 2017] Lecture 23:  Chapter 4.5 - proofs of 3 Sylow thms, remarks from the proofs. 

   -- Graded HW 5 with solution is returned

[Oct 16, 2017] Lecture 22:  Chapter 4.5 - ideas of 3 Sylow thms, two lemmas and proofs.  

   -- HW 6 (due on Oct. 23, Mon) is handed out.

[Oct 13, 2017] Lecture 21:  Chapter 4 - outline of topics in the chapter, a relationship between cyclic, abelian, solvable, non-abelian / Chapter 4.5 - statements of three Sylow theorems, Sylow-p-subgroup, n_p=1 <=> a Sylow-p-subgroup is normal in G.

[Oct 11, 2017] Lecture 20:  Chapters 3.4/3.5 - more arguments about subnormal, normal series, composition series, simple, solvable group, more examples, discussed: Cauchy Thm, |G|=p, prime, <=> Z_p <=>simple and abelian finite group, A_n with sgn, A_n, n >=5, is simple. 

   -- Graded HW 4 with solution is returned // An additional note for series of subgroups is haded-out.

[Oct 6, 2017] Lecture 19:  Chapter 3.4 - defined subnormal, normal series, composition series, examples, simple, solvable group. 

   -- HW 5 (due on Oct. 13, Fri) is handed out. // An additional note for series of subgroups is E-mailed.

[Oct 4, 2017] Lecture 18:  Chapter 3.3 - proofs of the 3 iso thms, remarks.

[Oct 2, 2017] Lecture 17:  Chapter 3.3 - a motivating example for isomorphism thms, introduced 3 iso thms and relevant remarks.

[Sep 29, 2017] Lecture 16:  Chapters 3.2-3.3 - Lagrange Theorem, examples, introduced the factor group, binary operation on the collection of left (right) cosets of H in G <=> H is normal in G, example for which H is not normal and the operation is not well-defined. 

   -- HW 4 (due on Oct. 6, Fri) is handed out.

[Sep 27, 2017] Lecture 15:  Chapters 3.1-3.2 - sketch of proofs of several equivalent arguments for being normal, recalled the definition/argument of equivalence relations, introduced left/right cosets, [G:H] = # of left cosets = # of right cosets. 

   -- Graded Exam 1 with solution is returned.

[Sep 25, 2017] Lecture 14: Chapter 2.5 - the lattice of subgroups, examples / Chapter 3 - introduced objecctives / Chapter 3.1 - recall def of homomorphism, properties, study kernel, definition of normal subgroup, equivalent conditions of being normal, examples, motive of cosets, factor groups

[Sep 22, 2017] Lecture 13: Exam 1 taken.

[Sep 20, 2017] Lecture 12: Chapter 2.3 - proved some of the main properties, discussed structure of cyclic groups.  

   -- Graded HW 3 with solution is returned.

[Sep 18, 2017] Lecture 11: Chapter 2.3 - defined cyclic groups, generator, examples, properties. 

   -- An additional note for cyclic groups is E-mailed.

[Sep 15, 2017] Lecture 10: Chapter 2.2 - examples related to inclusions {2} =< Z(G) =< C_G(A) =< N_G(A) =< G, defined commutator subgroup, stabilizer, examples, properties.

[Sep 13, 2017] Lecture 9: Chapter 2.2 - centralizer, normalizer, center, examples, their properties. / Chapter 2.4 - <A> the subgroup generated by a subset A in a group G, examples. 

   -- Graded HW 2 with solution is returned.

[Sep 11, 2017] Lecture 8: Chapter 2.1 - def of subgroups, examples, arguments related to subgroups, subgroup criterion and proof. / Chapter 2.2 - introduced the centralizer. 

   -- HW 3  (due on Sep. 18, Mon) is handed out.

[Sep 8, 2017] Lecture 7: Chapter 1.7 - definition, motivation of group actions, examples,  several terms of group actions.

[Sep 6, 2017] Lecture 6: Chapters 1.2-1.5 - introduced Q_8 - the quaternion group of order 8 / Chapter 1.6 - definitions of homomorphism, isomorphism, examples, properties. 

   -- Graded HW 1 with solution is returned. // HW 2 is distributed in a paper form.

[Sep 1, 2017] Lecture 5: Chapters 1.2-1.5 - some properties on S_n, even, odd, permutations, D_{2n}, its group presentation, examples. 

   -- HW 1 Due is extended // HW 2  (due on Sep. 11, Mon) is emailed. 

[Aug 30, 2017] Lecture 4: Chapters 1.2-1.5 - more about S_n, permutations, cycles, transpositions, orbits, examples, S_A with finite set A.

[Aug 28, 2017] Lecture 3: Chapter 1.1 - abelian groups, more properties of a group, order of an element in a group, order of a group / Chapters 1.2-1.5 - introduced S_n the symmetric group of degree n. 

[Aug 27, 2017] Office Hours Updated as above!

[Aug 25, 2017] Lecture 2: Chapter 1.1 - binary operation, associative, commutative, examples, Z, Q, R, C, Z_n. 

   -- HW 1 (due on Sep. 1, Fri) is distributed.

[Aug 23, 2017] Lecture 1: Syllabus distributed / listed course objectives and topics / Chapter 1.1 - definition of groups, binary operation, examples

[Aug 18, 2017] Syllabus is uploaded above in PDF.