S16-419

Introduction to Abstract Algebra II

Spring 2016

Instructor: Kwangho Choiy

Office: Neckers 283 (618-453-6508)
E-mail: kchoiy_at_siu_dot_edu
Office Hours: Tues11:30am-3:30pm/Thur 1:00pm-3:00pm, or by appointment. Emails are also available.

Course Website: https://sites.google.com/site/kchoiy/home/teaching/previous-courses/math-419

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

Textbook: 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, 45--47): ring of polynomials, ideals, factor rings, UFD, PID, Euclidean domain, 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 linked here, UPDATED on 1/10/2016.

Exams: There will be two mid-term exams and one final. No make-up exam will be accepted.

Exam 1 (in class on 2/18, THRS) covers Sections 22, 34--37.
Exam 2 (in class on 3/31, THRS) covers Sections 23, 26, 27, 29, 30.
Final Exam (10:15am-12:15pm on 5/10, TUES) will be comprehensive: approximately, 40% on all Sections in Exam 1-2, and 60% on Sections 31, 33, 48, 49, 50, 51, 53.

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.*

HW 1 (due on Jan. 28, Thur) -- updated on 1/26 (*ignore #5 and the questions in Section 35, which will be on HW 2 / read phi[G’] as phi[G] in #3-(c).)
HW 2 (due on Feb. 4, Thur) -- updated on 2/1 (*exclude b, d along with i in #17 of Section 35.)
HW 3 (due on Feb. 11, Thur) -- updated on 2/7 (*no need to submit "#1" "#2" and "#15 of Section 36".)
HW 4 (due on Mar. 1, Tue)
HW 5 (due on Mar. 8, Tue)
HW 6 (due on Mar. 24, Thr)
HW 7 (due on Apr. 12, Tue) -- Section 27 in #5 was corrected to Section 31.
HW 8 (due on Apr. 19, Tue)
HW 9 (due on Apr. 26, Tue) -- *no need to submit #2 - (3).
HW 10 (due on May 3, Tue) -- typo in #1 has been fixed and now {E:F} ≤ [E:F].

Updates and Remarks done during the semester- math419 - Spring 2016:

[May 5, 2016] Lecture 30: Section 51 - motivating example for separable extension, definitions of separable polynomial, extension and element, examples, more properties and remarks of separable extensions, perfect fields / Section 53 - state the main theorem of Galois theory, sketch of proof, example. Some hints and solutions for Practice Problems for Final emailed out!

[May 3, 2016] Lecture 29: Section 50 - examples, remarks of splitting fields, {E:F} = # G(E/F) when E is a splitting field over F, how to make Q(3rd root of 2) to be the splitting field of x^3-2, E is a splitting field over F <=> all auto of F bar fixing F becomes auto of E. Introduce a motivating example for separable extension. Practice Problems for Final distributed!

[Apr 28, 2016] Lecture 28: Section 48 - proved the conjugation isomorphism theorem, more examples, a brief motivation, statement of Galois main theorem. / Section 49 - stated the isomorphism extension theorem, its application, the splitting field of f(x).

[Apr 26, 2016] Lecture 27: Section 48 - state the conjugation isomorphism theorem, corollaries, applications, examples. HW 10 (due on May 3, Tue) distributed!

[Apr 21, 2016] Lecture 26: Section 48 - conjugate, G(E/F) - Aut(E) - the set of isomorphisms from E to a subfield if bar{E} (=the set of embeddings of E into bar{E}), conjugation theorem, if E = F(a_1, a_2, ... a_n) with ai are all zeros of an irreducible poly over F then G(E/F) is a subgroup of Sn (= the permutations on [n]). HW 8 returned with solution!

[Apr 19, 2016] Lecture 25: Section 48 - defined G(E/F) and gave examples, introduced E_sigma, the field in E fixed by sigma. HW 9 (due on Apr. 26, Tue) distributed!

[Apr 14, 2016] Lecture 24: Section 33 - more proofs and examples about finite extensions of a finite field. / Section 48 - introduce to Galois theory, motivation, examples, introduce to automorphisms, a group structure of Aut(E). HW 7 returned with solution!

[Apr 12, 2016] Lecture 23: Section 33 - more structure of finite fields, primitive roots of unity, examples. HW 8 (due on Apr. 19, Tue) distributed.

[Apr 7, 2016] Lecture 22: Section 31 - more properties of algebraic extension, algebraically closed field, algebraic closure of F / Section 33 - motivations, summary on finite extension fields.

[Apr 5, 2016] Lecture 21: Section 31 - def, examples, properties of algebraic extensions, introduced to algebraic closure of F in E. HW 7 (due on Apr. 12, Tue) distributed. Exam 2 returned with solution!

[Mar 31, 2016] Lecture 20: Exam 2.

[Mar 29, 2016] Lecture 19: Section 30 - motivation why to learn VS, def, examples, properties of VS, introduce, Span, linearly independence, basis, dimension, viewed F(alpha) as F-VS, deg(alpha, F)=dimF(alpha) for algebraic alpha over F. HW 6 returned with solution! Some hints and solutions for Practice Problems for EXAM 2 emailed out!

[Mar 24, 2016] Lecture 18: Section 29 - more new terms - algebraic, transcendental, irr(alpha, F), deg(alpha, F)..., theorems for algebraic, transcendental extensions, a connection between the notions of an extension field and a vector space. Practice Problems for Exam 2 distributed!

[Mar 22, 2016] Lecture 17: Section 29 - new terms to start the field theory, extension fields, base fields, stated and proved the existence of extension fields, examples.

[Mar 10, 2016] Lecture 16: Section 27 - introduce principal ideals, more about maximal and prime ideals, more properties of F[x] related to ideals. HW 5 returned with solution!

[Mar 8, 2016] Lecture 15: Section 27 - def of maximal, prime ideals, examples, theorems in comm with 1 related to prime and maximal ideals. HW 6 (due on Mar. 24, Thr) distributed!

[Mar 3, 2016] Lecture 14: Section 26 - factor ring, fundamental theorem of a ring homomorphism, more about ideals. HW 4 returned with solution!

[Mar 1, 2016] Lecture 13: Section 26 - motivation to define ideals, properties of ring homomorphisms, study kernel of ring homomorphism, definition of ideals, in the set of additive (left) cosets of a subring H in a ring R, the multiplication (a+H)(b+H) = ab +H is well-defined iff H is an ideal. HW 5 (due on Mar. 8, Tue) distributed.

[Feb 25, 2016] Lecture 12: Section 23 - definition of irreducible, reducible polynomial, two criteria for determination of irreducibility: a typical, Eisenstein criterion, unique factorization, F* is cyclic when F is finite.

[Feb 23, 2016] Lecture 11: Section 23 - motivation to learn F[x],division algorithm in F[x], introduced notions of irreducible, unique factorization. HW 4 (due on Mar. 1, Tue) distributed. Exam 1 returned with solution!

[Feb 18, 2016] Lecture 10: Exam 1.

[Feb 16, 2016] Lecture 9: Section 22 - introduce R[x], evaluation homomorphism, kernel of a ring homomorphism. / discussions on Prac Prob for Exam 1. HW 3 returned with solution!

[Feb 11, 2016] Lecture 8: Section 36-37 - define normalizer and examples / Section 22 - review definition, basic properties of rings, examples. Practice Problems for Exam 1 distributed!

[Feb 9, 2016] Lecture 7: Section 36-37 - more applications of Sylow theorems, recall def, properties of a group action and discussed class equation, proof of Cauchy's theorem. HW 2 returned with solution!

[Feb 4, 2016] Lecture 6: Section 36-37 - statement of 3 Sylow theorems, examples, applications. HW 3 (due on Feb. 11, Thur) distributed!

[Feb 2, 2016] Lecture 5: Section 35 - more examples for solvable groups / Section 36-37 - prelim for Sylow theorems: motivating examples, definitions of p-group, p-subgroup, Sylow p-subgroup, examples, Cauchy's theorem, corollary to Cauchy's thm about p-group. HW 1 returned with solution!

[Jan 28, 2016] Lecture 4: Section 34 / Section 35 -- application of three isomorphism theorems, introduced HvN / introduced subnormal series, composition series, solvable groups, gave their examples. HW 2 (due on Feb. 4, Thur) distributed!

[Jan 26, 2016] Lecture 3: Section 34 -- proved three isomorphism theorems, stated an application of the 1st isomorphism theorem.

[Jan 21, 2016] Lecture 2: Section 34 -- stated three isomorphism theorems and gave examples and necessary arguments. HW 1 (due on Jan. 28, Thur) distributed!

[Jan 19, 2016] Lecture 1: Syllabus / Section 34 -- discussed two motivating examples for study of advanced group theory, and recalled some notions from Math 319 for three isomorphism theorems: def of a group, normal subgroups, homomorphism, isomorphism, the kernel, S_n, A_n, transpositions, even/odd permutations, Z_n, Z/nZ.

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