17Sp-419

Introduction to Abstract Algebra II

Spring 2017

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu 

• Office Hours: T 1-5pm / W 3:30-4:30pm / Th 2-3pm, or by appointment. Emails are also available.

Course Website: https://sites.google.com/site/kchoiy/home/teaching/17sp-419

Class Meeting: TTh 10:00am - 11:15am in EGRA 310

Textbook/Topics: A First Course in Abstract Algebra 7th edition by John B. Fraleigh. The topics include:

• advanced group theory (Sections 34--37) : isomorphism theorems, series of groups, solvable groups, Sylow theorems, etc. 

• rings and ideals (Sections 22, 23, 26, 27): ring of polynomials, ideals, factor rings, integral domains, etc. 

• field theory (Sections 29--33, 48-56): field extensions, finite fields, Galois theory, insolvability of the quintic, impossibility of trisecting an angle, etc.

Syllabus / Course Schedule: It is required to read carefully our syllabus and schedule [PDF] - updated on 3/9/2017.

Exams: There will be two mid-term exams and one final. No make-up exam will be accepted. The following coverage and schedule may be subject to change, but they will be confirmed at least one week ahead of time:  

1.  Exam 1 (in class on 2/16, THUR) covers Sections 34–37.

2.  Exam 2 (in class on 3/30, THUR) covers Sections 22, 23, 26, 27, 29 - announced in class on 3/23.

3.  Final Exam (10:15am-12:15pm on 5/9, TUES) will be will be comprehensive: approximately, 40% on Sections 22, 23, 26, 27, 29, 34-37 (Exam 1 &2 coverage), and 60% on Sections 30, 31, 33, 48-51, 53. - announced on 4/27 in class and by email.

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.*

1. HW 1 (due on Jan. 26, Thur) - typo: #4(b), N is normal should be "H is normal"

2. HW 2 (due on Feb. 2, Thur)

3. HW 3 (due on Feb. 9, Thur)

4. HW 4 (due on Feb. 28, Tues) - update: due is now on Mar. 2, Thursday.

5. HW 5 (due on Mar. 9, Thur)

6. HW 6 (due on Mar. 23, Thur)

7. HW 7 (due on Apr. 11, Tues) 

8. HW 8 (due on Apr. 18, Tues)

9. HW 9 (due on Apr. 25, Tues) 

10. HW 10 (due on May 2, Tues)

Updates and Remarks - MATH419 - Spring 2017:

[Mar 4, 2017] Lecture 30: Section 50 - more examples for splitting / Section 51 - definition of separable, examples, / Section 49 - definition of {E:F},  relations between {E:F}, [E:F], and |G(E/F)|, examples, several remarks about splitting, separable.

[Mar 2, 2017] Lecture 29: Section 53 - main theorem of Galois theory, properties, example / Section 50 - introduced splitting fields, examples. Solution to HW 10 is distributed.

[Apr 27, 2017] Lecture 28: Section 48 - def of conjugations, the conjugation theorem, examples, how to compute Aut_F(E), Fronebius map on finite field. Graded HW 9 with solution is returned/ Final exam (5/9) is announced / practice problems are distributed.

[Apr 25, 2017] Lecture 27: Section 48 - Aut(E), Aut_F(E), group action of Aut(E) on E, the fixed subfield in E, a subgroup in Aut_F(E), outlines toward Galois theorem. HW 10(due on 5/2, Tues) is handed out. 

[Apr 20, 2017] Lecture 26: Section 48 - more details about introduction to Galois theory, concrete setting up for its connection with group action / primitive n-th roots of 1 from Section 33 and its connection to group theory. Graded HW 8 with solution is returned.

[Apr 18, 2017] Lecture 25: Section 33 - any finite field is of order p^n, uniqueness of GF(p^n) up to isomorphism, examples / Section 48 - motivation of Galois theory, its connection with group action. HW 9(due on 4/25, Tues) is handed out / Two additional notes (4th) are emailed out.

[Apr 13, 2017] Lecture 24: Section 33 - characterization of finite fields, properties, existence of GF(p^n), examples. Graded HW 7 with solution is returned

[Apr 11, 2017] Lecture 23: Section 31 - algebraic extension, finite extension, finite=>algebraic, algebraic closure, algerbaically closed field, examples. HW 8(due on 4/18, Tues) is emailed out.

[Apr 6, 2017] Lecture 22: Section 30 - motivations of vector spaces, connection with vector spaces and algebraic over F, def of vector space, properties, Span(S), lin ind, dim, basis for F(alpha) over F with algebraic alpha over F.

[Apr 4, 2017] Lecture 21: Review Exam 2 / Section 29 - Kronecker's Theorem, sketch of proof, examples.  Graded EXAM 2 with solution is returned / HW 7(due on 4/11, Tues) is emailed out.

[Mar 30, 2017] Lecture 20: Exam 2

[Mar 28, 2017] Lecture 19: Section 29 - irr(alpha, F), deg(alpha, F), important theorems/properties regarding algebraic over F and transcendental over F, examples. Graded HW 6 with solution is returned / Some solutions for practice problems are emailed out.

[Mar 23, 2017] Lecture 18: Section 27 - prime, maximal ideals in F[x] / Section 29 - motivation, definition of algebraic over F and transcendental over F. Exam 2(3/30) is announced / practice problems are distributed.

[Mar 21, 2017] Lecture 17: Section 27 - prime, maximal, principal ideals, examples, all ideals in a field, prime <=> R/N integral domain, maximal <=> R/N field, sketch of proofs. Graded HW 5 with solution is returned 

[Mar 9, 2017] Lecture 16: Section 26 - ring homomorphism, kernel, ideal, their properties, factor ring, examples. Graded HW 4 with solution is returned / HW 6(due on 3/23, Thur) is handed out / An additional note (3rd) is emailed out. Office hours will be changed from 3/20 on as written above.

[Mar 7, 2017] Lecture 15: Section 23 - irreducible polynomial, how to determine the reducibility, Eisenstein criterion, the uniqueness of factorization of F[x], the structure of F -{0} when F is a finite field. An additional note (2nd) is emailed out.

[Mar 3, 2017] An additional note (1st) is emailed out.

[Mar 2, 2017] Lecture 14: Section 23 - more computations on R[x], three major properties of F[x], proof of division algorithm of F[x]. HW 5(due on 3/9, Thur) is handed out.

[Feb 28, 2017] Lecture 13: Recalled notions on integral domains and fields / Section 22 - ring homomorphism, evaluation homomorphism, examples. 

[Feb 23, 2017] Lecture 12: Recalled basic notions of rings from Sections 18, 19, 21.

[Feb 21, 2017] Lecture 11: Reviewed and discussed Exam 1. Graded EXAM 1 with solution is returned / HW 4(due on 3/2, Thur) is handed out.

[Feb 16, 2017] Lecture 10: Exam 1

[Feb 14, 2017] Lecture 9: more about group actions, more examples, class equation, some useful theorems / review for EXAM 1. Graded HW 3 is returned with solution.

[Feb 9, 2017] Lecture 8: Review of group actions, definitions, examples, G_x, Gx, |Gx|=[G:G_x], conjugate action, centralizer, normalizer. Exam 1(2/16) is announced / practice problems are distributed.

[Feb 7, 2017] Lecture 7: Section 37 - applications of Sylow theorems, some useful arguments. Graded HW 2 is returned with solution.

[Feb 2, 2017] Lecture 6: Section 36/37 - 2nd and 3rd Sylow theorems, examples, applications of Sylow theorems. HW 3(due on 2/9, Thur) is handed out.

[Jan 31, 2017] Lecture 5: Section 35 - one further refinement for solvable groups / Section 36 - p-groups, some arguments related to p-groups, 1st sylow theorem, examples. Graded HW 1 is returned with solution.

[Jan 26, 2017] Lecture 4: Section 35 - simple groups, composition series, Joran-Hölder Theorem, solvable groups, examples. HW 2(due on 2/2, Thur) is handed out.

[Jan 24, 2017] Lecture 3: Section 34 - proof of 3rd isom theorem / Section 35 - def and examples of subnormal and normal series.

[Jan 19, 2017] Lecture 2: Section 34 - proof of 1st, 2nd isom theorems, remarks and examples, joint of H and K. HW 1(due on 1/26, Thur) is handed out.

[Jan 17, 2017] Lecture 1: Intro and outline of the course / reviewed some group theory (319) / Section 34 - stated 3 isomorphism theorems. Syllabus distributed.

[Jan 6, 2017] Syllabus/tentative Schedule are uploaded above in PDF.