Math419_S19

Introduction to Abstract Algebra II

Spring 2019

Instructor: Kwangho Choiy

• Office: Neckers 283 (618-453-6508)

• E-mail: kchoiy_at_siu_dot_edu 

• Office Hours: T 9-10am; 11:15-12pm; 1:15-3pm / Th 9:15-10am; 11:15-12pm; 2-3pm, or by appointment. Emails are also available.

Course Website: https://sites.google.com/site/kchoiy/teaching/math419_s19

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 [here] - updated on 1/10/2019.

Exams: There will be one mid-term 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.  Mid-term (in class , 10:00am - 11:15am on 3/7, Thur) covers Sections 34-37, 22, 23, 26 (7 sections)

2.  Final (8-10am on May 10th, Fri) covers Sections 27, 29--31, 33, 48-51, 53 (10 sections) - announced in class & by emails & D2L on 4/30.

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 24) -- updated HW 1 emailed after class on 1/17.

2. HW 2 (due on Jan 31)

3. HW 3 (due on Feb 7)

4. HW 4 (due on Feb 14)

5. HW 5 (due on Feb 21) **No need to submit #6&#17 in Sec 23

6. HW 6 (due on Feb 28)  

7. HW 7 (due on Mar 28)

8. HW 8 (due on Apr 4)

9. HW 9 (due on Apr 11)

10. HW 10 (due on Apr 18)

11. HW 11 (no need to submit)

Updates and Remarks - MATH419 - Spring 2019:

[May 2, Thu] Sec 48-56 - Fundamental theorem for Galois theory, examples, {E:F}, application of Galois Thy emailed out.

[Apr 30, Tue] Sec 48-56 -two more motivating examples, accordingly, defined splitting field, separable field, examples.

[Apr 25, Thu] Sec 48-56 - Using G_x, X_g, defined (Aut_F(E))_K and E_H, example, brief glance for Fundamental theorem for Galois theory.

[Apr 23, Tue] Sec 48-56 - G_x, X_g, G_{subset}, examples for Aut_F(E), new terms. Graded HW 10 with solution is returned.

[Apr 18, Thu] Sec 48-56 - set up Aut_F(E) action on E, three major questions for Galois theory, proved Aut(E) is a group and Aut_F(E) < Aut(E), examples, two crucial conclusions for a technique to compute Aut_F(E). HW 11 (no need to submit) is distributed.

[Apr 16, Tue] Sec 48-56 - recalled group actions, G_x, Gx, X_g, connection between group action and a group homo G to S_X, examples. Graded HW 9 & Quiz 9 with solutions are returned.

[Apr 11, Thu] Sec 33 - further arguments related to finite fields, examples, irreducible poly for any degree over Z_p / Sec 48 - motivation, history about Galois theory, big picture, recalled group action. HW 10 (due on Apr 18) is distributed / Quiz 9 is taken.

[Apr 9, Tue] Sec 33 - several description of finite fields,  any finite field is of order p^n, existence of GF(p^n), uniqueness of GF(p^n) up to isomorphism, examples. Graded HW 8 & Quiz 8 with solutions are returned.

[Apr 4, Thu] Sec 33 - Char(a commutative ring with 1), Char(F)=0, p, Char(finite fields)=p, Zp embedding into any finite field, examples. HW 9 (due on Apr 11) is distributed / Quiz 8 is taken.

[Apr 2, Tue] Sec 31 - proof of statement of finite extension => algebraic, counterexample for its converse, more arguments and examples for finite extensions, basic backgrounds for vector spaces from Sec 30, algebraic closure, algebraically closed field, several examples and arguments. Graded HW 7 & Quiz 7 with solutions are returned.

[Mar 28, Thu] Sec 29 - Kronecker theorem, existence of extension fields, description of F[alpha] with alg alpha/F, examples / Sec 31 - introduced new terms, algebraic extensions, [E:F], finite extension, statement of finite extension => algebraic. HW 8 (due on Apr 4) is distributed / Quiz 7 is taken.

[Mar 26, Tue] Sec 29 - properties of algebraic/F and transcendental/F, F[alpha], F(alpha), irr(alpha,F), deg(alpha,F), concrete description of F[alpha]=F(alpha) for algebraic alpha, examples, rough concept about how to construct an extension field from a given base field.

[Mar 21, Thu] Sec 27 - proved two important properties of F[x] / Sec 29 - motivation, introduced algebraic, transcendental, examples, introduced how to construct extension fields using evaluation maps. HW 7 (due on Mar 28) is distributed.

[Mar 19, Tue] Reviewed Mid-term / Sec 27 - proof of comm with 1, N prime ideal <=> R/N ID, introduced two important properties of F[x]. Returned graded Mid-term with solution.

[Mar 7, Thu] Mid-term taken.

[Mar 5, Tue] Sec 27 - prime => max, an example for prime ideal not maximal / Review for Mid-Term - summarized topics in the coverage. Graded HW 6 & Quiz 6 with solutions are returned.

[Feb 28, Thu] Sec 27 - proof of comm with 1, N max ideal <=> R/N field, def/proof about principal ideal, def/example of prime ideals, stated comm with 1, N prime ideal <=> R/N I.D. Quiz 6 is taken / Mid-term announced in class & by E-mail.

[Feb 26, Tue] Sec 26 - more arguments, examples about factor rings, proof of (r+N)(s+N)=rs+N is well-defined iff N is ideal / Sec 27 - motive, definition of maximal ideals, examples. Graded HW 5 & Quiz 5 with solutions are returned.

[Feb 21, Thu] Sec 26 - further properties of kernel, definition, example of ideals, an example of subring but not ideal, two special cases of ideals, factor ring, well-definedness of two binary operations, (r+N)(s+N)=rs+N is well-defined iff N is ideal, example. HW 6 (due on Feb 28) is distributed / Quiz 5 is taken.

[Feb 19, Tue] Sec 23 - reducibility in F[x], Eisenstein criterion, examples / Sec 26 - definition, properties, example of ring homomorphism, kernel, properties of kernel. Graded HW 4 & Quiz 4 with solutions are returned.

[Feb 14, Thu] Sec 23 - proof of existence, uniqueness in division algorithm for F[x], example, techniques for division in F[x], unique factorization in F[x], remark. HW 5 (due on Feb 21) is distributed / Quiz 4 is taken.

[Feb 12, Tue] Sec 23 - R is comm with 1(I.D.) => R[x] is comm with 1 (I.D.), F[x] is I.D., three properties of F[x] corresponding to division algorithm, prime number, unique prime factorization in Z, respectively, statement of division algorithm in F[x], definition of irreducible polynomial in F[x]. Graded HW 3 & Quiz 3 with solutions are returned.

[Feb 7, Thu] Sec 22 - more reviews from 319 including cautions related to rings, unity 1, units, commutative rings, zero divisors, integral domains, fields, division rings, examples, diagram consisting of all kinds of rings, ring homomorphism, R[x], \phi_\alpha: R[x] -> R. An additional note is E-mailed / HW 4 (due on Feb 14) is distributed / Quiz 3 is taken.

[Feb 5, Tue] Sec 36-37 - summarize 3 Sylow theorems and applications, proofs / Sec 22 - review of definition of rings, examples. Graded HW 2 & Quiz 2 with solutions are returned.

[Jan 31, Thu] Sec 36-37 - examples for 1st Sylow, stated 2nd, 3rd Sylow, examples, applications. Two additional notes are distributed / HW 3 (due on Feb 7) is distributed / Quiz 2 is taken.

[Jan 29, Tue] Sec 35 - composition series, how to show not-solvable, examples / Sec 36-37 - recalled some useful notions and statements from 319, stated 1st Sylow thm, examples, p-group, p-subgroup, Sylow p-subgroup, examples. Graded HW 1 & Quiz 1 with solutions are returned with some comments and reviews.

[Jan 24, Thu] Sec 35 - normal series, subnormal series, simple groups, solvable groups, examples, remarks. HW 2 (due on Jan 31) is distributed / Quiz 1 is taken.

[Jan 22, Tue] Sec 34 - remarks and applications of 1st isom thm, proof and examples of 2nd, 3rd isomorphism thms.

[Jan 17, Thu] Sec 34 - proof, examples of 1st isomorphism thm, a crucial lemma. HW 1 (due on Jan 24) is handed out.

[Jan 15, Tue] syllabus distributed, main objectives Sec 34 - 1st isomorphism theorem, recalled several examples and notions from 319 including isomorphism, kernel, normal subgroups, cosets, factor groups, S_n, A_n.