MATH:5010 - Abstract Algebra II, Spring 2017

This will be updated after each lecture.

Book references: Dummit and Foote (D-F), Lang's "Algebra" book (Dummit and Foote has been put on reserve in the media/reserve area at the south end of the first floor of the Main Library. Excerpts from Lang's "Algebra" book will be put on ICON when we come to field theory.)

Monday	Wednesday	Friday
Jan. 16 MARTIN LUTHER KING DAY	Jan. 18 In Lecture: Review of structure of module homomorphisms Review of quotient modules Review of isomorphism theorems Generation of modules Book reference: D-F: 10.2, 10.3 Read the SYLLABUS. HOMEWORK 1 Part A Problems: due on Tu, 01/24. Part B Problems: due on Fr, 01/27.	Jan. 20 In Lecture: Direct products and direct sums of modules Free modules Book reference: D-F: 10.3
TUESDAY, JAN. 17, 11:00-11:50 in 118 MH MODULE THEORY Erik will start the review: Review of modules Review of basic definitions and examples of modules Review of algebras (over a commutative ring) Review of module homomorphisms Book reference: D-F: 10.1, 10.2
Jan. 23 In Lecture: Free modules Tensor products (general construction) Book reference: D-F: 10.3, 10.4 (pp. 364-366, examples (1)-(5) on pp. 368-369)	Jan. 25 In Lecture: Tensor products Exact sequences of modules Book reference: D-F: 10.4 (pp. 364-366, examples (1)-(5) on pp. 368-369), 10.5 (pp. 378-385) HOMEWORK 2 Part A Problems: due on Tu, 01/31. Part B Problems: due on Fr, 02/03.	Jan. 27 In Lecture: The Short Five Lemma Split exact sequences of modules Exact sequences ending in free modules split Basics of vector spaces Book reference: D-F: 10.5 (pp. 378-385), 11.1
Jan. 30 In Lecture: Basics of vector spaces Matrices Book reference: D-F: 11.1, 11.2	Feb. 01 In Lecture: Matrices Change of basis Book reference: D-F: 11.2 HOMEWORK 3 Part A Problems: due on Tu, 02/07. Part B Problems: due on Fr, 02/10.	Feb. 03 In Lecture: Dual vector spaces Book reference: D-F: 11.3
Feb. 06 In Lecture: Multilinear functions Determinants Book reference: D-F: 11.4	Feb. 08 In Lecture: Determinants Noetherian rings and modules Book reference: D-F: 11.4, 12.1 HOMEWORK 4 Part A Problems: due on Tu, 02/14. Part B Problems: due on Fr, 02/17.	Feb. 10 In Lecture: Noetherian rings and modules Rank of an R-module when R is an integral domain Finitely generated free modules over PIDs Book reference: D-F: 12.1
Feb. 13 In Lecture: Finitely generated free modules over PIDs Fundamental theorem of finitely generated modules over PIDs: Invariant Factor form Book reference: D-F: 12.1	Feb. 15 In Lecture: Fundamental theorem of finitely generated modules over PIDs: Invariant Factor form and Elementary Divisor form Book reference: D-F: 12.1 HOMEWORK 5 Part A Problems: due on Tu, 02/21. Part B Problems: due on Fr, 02/24.	Feb. 17 In Lecture: Completing the proof of the uniqueness of the fundamental theorems Applying the fundamental theorems to matrices Eigenvectors and eigenvalues of linear transformations and square matrices Characteristic polynomial χA(x) and minimal polynomial mA(x) of a linear transformation and a square matrix A Book reference: D-F: 12.2
Feb. 20 In Lecture: The rational canonical form (invariant factor form) The Cayley-Hamilton Theorem Book reference: D-F: 12.2	Feb. 22 In Lecture: How to compute rational canonical forms: Invariant Factor Decomposition Algorithm The Jordan canonical form (elementary divisor form) Book reference: D-F: 12.2, 12.3	Feb. 24 In Lecture: The Jordan canonical form Diagonal matrices Book reference: D-F: 12.3
Feb. 27 In Lecture: How to compute Jordan canonical forms: Elementary Divisor Decomposition Algorithm Book reference: D-F: 12.3	Mar. 01 FIELD THEORY In Lecture: Characteristic of a field Field extensions, algebraic extensions Minimal polynomial mα,F(x) (Lang calls this the irreducible polynomial Irr(α,F,x)) Book reference: D-F: 13.1, 13.2; also, Lang's "Algebra" book: V.1 (posted on ICON on the MATH:5010:0AAA site in "Modules") HOMEWORK 6 Part A Problems: due on Tu, 03/07. Part B Problems: due on Fr, 03/10.	Mar. 03 In Lecture: If p(x) ∈ F[x] of degree ≥ 1 then there exists a finite extension E/F in which p(x) has a root Embeddings, extensions of embeddings Book reference D-F: 13.1, 13.2; also, Lang's "Algebra" book: V.1
Mar. 06 In Lecture: Multiplicativity of degrees Finitely generated field extensions, simple extensions Towers of algebraic extensions Book reference: D-F: 13.1, 13.2; Lang's "Algebra" book V.1	Mar. 08 In Lecture: Composita Algebraically closed fields Existence of algebraically closed fields Book reference: D-F: 13.2; Lang's "Algebra" book V.2 HOMEWORK 7 Part A Problems: due on Tu, 03/21. Part B Problems: due on Fr, 03/24.	Mar. 10 In Lecture: Existence of algebraically closed fields More on extensions of embeddings Uniqueness of algebraic closures Book reference: Lang's "Algebra" book V.2
Mar. 13 SPRING BREAK	Mar. 15 SPRING BREAK	Mar. 17 SPRING BREAK
Mar. 20 In Lecture: Uniqueness of algebraic closures Splitting fields Book reference: Lang's "Algebra" book V.2, V.3	Mar. 22 In Lecture: More on splitting fields Normal extensions Separable and inseparable polynomials Book reference: Lang's "Algebra" book V.3; D-F: 13.5 HOMEWORK 8 Part A Problems: due on Tu, 03/28. Part B Problems: due on Fr, 03/31.	Mar. 24 In Lecture: Separable and inseparable polynomials Finite fields, Frobenius endomorphism Book reference: D-F: 13.5
Mar. 27 In Lecture: Perfect fields Separable extensions Book reference: D-F: 13.5	Mar. 29 In Lecture: Finite separable extensions - degree versus number of embeddings Primitive element theorem Book reference: Lang's "Algebra" book: V.4 (simplified results; simpler version of Theorem V.4.6) HOMEWORK 9 Part A Problems: due on Tu, 04/04. Part B problems: due on Fr, 04/07.	Mar. 31 In Lecture: Cyclotomic polynomials and extensions Book reference: D-F: 13.6
Apr. 03 In Lecture: Introduction to Galois theory IMPORTANT: We will define a field extension K/F to be Galois if K/F is separable and normal. This definition works for finite and infinite extensions! (The definition in D-F only works for finite extensions.) If K/F is finite Galois then |Gal(K/F)| = [K:F] Book reference: D-F: 14.1; Lang's "Algebra" book VI.1	Apr. 05 In Lecture: The fundamental theorem of Galois theory Examples of Galois groups Book reference: D-F: 14.2; Lang's "Algebra" book VI.1	Apr. 07 In Lecture: Examples of Galois groups Galois conjugates Proof of the fundamental theorem of Galois theory Book reference: D-F: 14.2; Lang's "Algebra" book VI.1
Apr. 10 In Lecture: Proof of the fundamental theorem of Galois theory: Artin's Theorem Book reference: Lang's "Algebra" book VI.1 (D-F 14.2)	Apr. 12 In Lecture: Completion of the proof of the fundamental theorem of Galois theory Abelian/cyclic (Galois) extensions Galois groups of cyclotomic extensions Book reference: Lang's "Algebra" book VI.1 (D-F 14.2); D-F: 14.5 (Theorem 26) HOMEWORK 10 Part A Problems: due on Tu, 04/18. Part B problems: due on Fr, 04/21.	Apr. 14 In Lecture: Galois groups over finite fields Composite extensions Book reference: D-F: 14.3, 14.4
Apr. 17 DISCUSSION SESSION DURING LECTURE in 105 MLH (see Announcements)	Apr. 19 In Lecture: Linear independence of characters Cyclic extensions Book reference: D-F: 14.2 (p. 569), 14.7 HOMEWORK 11 Part A Problems: due on Tu, 04/25. Part B Problems: due on Fr, 04/28 (PLEASE PUT THE B PROBLEMS INTO ERIK'S MAILBOX BY 12:30PM on 04/28).	Apr. 21 REGULAR LECTURE 11:30-12:20PM in 105 MLH: In Lecture: Cyclic extensions Root extensions, solvability by radicals (assume char(F)=0) Book reference: D-F: 14.7 EXTRA LECTURE 5:30-6:20PM in 218 MLH: (see Announcements) In Lecture: A polynomial in F[x] is solvable by radicals iff the Galois group of its splitting field is solvable (assume char(F)=0) An example of a polynomial of degree 5 that is not solvable by radicals over Q Book reference: D-F: 14.7
LECTURE IN DISCUSSION SESSION TUESDAY, APR. 18 (11:00-11:50 in 118 MH) (see Announcements) Galois closure Review of solvable groups (what we proved last semester plus a criterion for finite groups) Book reference: D-F: 14.4; (Solvable groups D-F: p. 105 and p. 195/6)
Apr. 24 In Lecture: A criterion for an irreducible polynomial of prime degree p in Q[x] to have Galois group isomorphic to Sp The general polynomial of degree n Discriminants Book reference: D-F: 14.7, 14.6	Apr. 26 In Lecture: Discriminants How to construct finite abelian extensions Book reference: D-F: 14.6, 14.5 (Corollary 28) Some extra problems (not to be handed in)	Apr. 28 NO LECTURE The lecture was made up on Friday, April 21, 5:30-6:20pm, in 218 MLH. (PLEASE PUT THE B PROBLEMS OF HOMEWORK 11 INTO ERIK'S MAILBOX BY 12:30PM on 04/28).
May 01 In Lecture: Constructions with straightedge and compass Book reference: D-F: 13.3	May 03 In Lecture: Constructions with straightedge and compass: Classical Greek problems Book reference: D-F: 13.3	May 05 In Lecture: Constructions with straightedge and compass: Connection to Galois theory Book reference D-F: 13.3

This will be updated after each lecture.

Book references: Dummit and Foote (D-F), Lang's "Algebra" book (Dummit and Foote has been put on reserve in the media/reserve area at the south end of the first floor of the Main Library. Excerpts from Lang's "Algebra" book can be found on ICON.)