Algebra I

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Algebra I
Fall 2012

Office hour at the graduate center: Tue 3:30-4:30 in 4214-11

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Class Calendar

Problem Set 1. Due at the start of class on Tue, Sept 11.

Problem Set 2. Due at the start of class on Thur, Sept 20.

No class on Tue, Sept 18 or Tue, Sept 25.

Problem Set 3.

Problem Set 4. with material on cyclic, symmetric groups. Due Oct 9.

Problem Set 5. Due at the start of class on Thur, Oct 18. Solutions to Set 5

Problem Set 6.

Problem Set 7. Due at the start of class on Thur, Nov 15.

Problem Set 8. Due at the start of class on Tue, Dec 4.

To make up for the two classes missed due to the hurricane there will be a class on Thur, Dec 13 and some extra handouts: Tensor products, Fields

Problem Set 9.

The take-home final will be given out at the end of class on Thur, Dec 13. Return to me in my office 4214-11 between 2 and 3 pm on Thur, Dec 20 (or email it to me by that time). Here it is - pdf, tex.

Algebra continues next semester with Prof. K Kramer.

1 Aug 28 Sections 1.1, 1.2, 1.3, 1.5, 1.6 of the main text, Dummit and Foote.

2 Aug 30 Sections 1.7, 2.1, 2.2

3 Sept 4 Sections 2.4, 2.5, 3.1

4 Sept 6 Sections 3.2, 3.3 with proofs of the isomorphism theorems. Definitions of rings, fields, modules.

5 Sept 11 Sections 3.4, 3.5, 4.1, 4.2, 4.3 with the class equation

6 Sept 13 Sections 4.5 with proofs of the Sylow theorems and the proof from Lang that A_n is simple for all n at least 5.

7 Sept 20 Problem set 1 returned. Rigid motions of the cube, automorphisms of Q, applications of Sylow theorems, Sections 5.1, 5.2.

8 Sept 27 Problem set 2 returned. Sections 5.2, 5.5.

9 Oct 2 Sections 5.2, 5.3: material on cyclic groups, partitions, counting, see problem set 4.

10 Oct 4 Section 6.1: solvable, nilpotent groups, proof that S_n not solvable for all n at least 5.

11 Oct 9 Section 6.3: free groups.

12 Oct 11 Sections 7.1, 7.2: introduction to rings.

13 Oct 16 Sections 7.3, 7.4: ideals.

14 Oct 18 Problem set 4 returned. Sections 7.4, 7.5: maximal, prime ideals, rings and fields of fractions.

15 Oct 23 Sections 7.6, 8.1: Chinese remainder theorem, Euclidean Domains.

16 Oct 25 Sections 8.2, 8.3: P.I.D.s, Unique factorization.

Oct 30 CUNY classes cancelled

Nov 1 CUNY classes cancelled

17 Nov 6 Problem set 5 returned with solutions. Sections 9.1, 9.2, 9.3: Gauss' Lemma, Unique factorization in polynomial rings.

18 Nov 8 Sections 9.4, 9.5: irreducibility criteria, constructing finite fields.

19 Nov 13 Sections 9.5, 9.6: finite multiplicative subgroups of field cyclic, Hilbert's Basis Theorem.

20 Nov 15 Sections 10.1, 10.2: introduction to modules.

21 Nov 20 Sections 10.3, 10.4: free modules, extension of scalars.

22 Nov 27 Problem set 7 returned. Section 10.4: constructing general tensor products, universal properties, examples.

23 Extra handouts: Tensor products, Fields

24 Nov 29 Sections 10.4, 10.5: properties of tensor products, tensor products of free modules, exact sequences.

25 Dec 4 Section 10.5: split exact sequences, projective modules and their properties.

26 Dec 6 Section 10.5: injective and flat modules. Section 11.1: dimension of vector space well defined.

27 Dec 11 Section 11.2: rank, nullity, matrix of a linear transformation.

28 Dec 13 Sections 11.3, 11.4: dual space, determinants. Take-home final given out - pdf, tex.

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1	Aug 28	Sections 1.1, 1.2, 1.3, 1.5, 1.6 of the main text, Dummit and Foote.
2	Aug 30	Sections 1.7, 2.1, 2.2
3	Sept 4	Sections 2.4, 2.5, 3.1
4	Sept 6	Sections 3.2, 3.3 with proofs of the isomorphism theorems. Definitions of rings, fields, modules.
5	Sept 11	Sections 3.4, 3.5, 4.1, 4.2, 4.3 with the class equation
6	Sept 13	Sections 4.5 with proofs of the Sylow theorems and the proof from Lang that A_n is simple for all n at least 5.
7	Sept 20	Problem set 1 returned. Rigid motions of the cube, automorphisms of Q, applications of Sylow theorems, Sections 5.1, 5.2.
8	Sept 27	Problem set 2 returned. Sections 5.2, 5.5.
9	Oct 2	Sections 5.2, 5.3: material on cyclic groups, partitions, counting, see problem set 4.
10	Oct 4	Section 6.1: solvable, nilpotent groups, proof that S_n not solvable for all n at least 5.
11	Oct 9	Section 6.3: free groups.
12	Oct 11	Sections 7.1, 7.2: introduction to rings.
13	Oct 16	Sections 7.3, 7.4: ideals.
14	Oct 18	Problem set 4 returned. Sections 7.4, 7.5: maximal, prime ideals, rings and fields of fractions.
15	Oct 23	Sections 7.6, 8.1: Chinese remainder theorem, Euclidean Domains.
16	Oct 25	Sections 8.2, 8.3: P.I.D.s, Unique factorization.
	Oct 30	CUNY classes cancelled
	Nov 1	CUNY classes cancelled
17	Nov 6	Problem set 5 returned with solutions. Sections 9.1, 9.2, 9.3: Gauss' Lemma, Unique factorization in polynomial rings.
18	Nov 8	Sections 9.4, 9.5: irreducibility criteria, constructing finite fields.
19	Nov 13	Sections 9.5, 9.6: finite multiplicative subgroups of field cyclic, Hilbert's Basis Theorem.
20	Nov 15	Sections 10.1, 10.2: introduction to modules.
21	Nov 20	Sections 10.3, 10.4: free modules, extension of scalars.
22	Nov 27	Problem set 7 returned. Section 10.4: constructing general tensor products, universal properties, examples.
23		Extra handouts: Tensor products, Fields
24	Nov 29	Sections 10.4, 10.5: properties of tensor products, tensor products of free modules, exact sequences.
25	Dec 4	Section 10.5: split exact sequences, projective modules and their properties.
26	Dec 6	Section 10.5: injective and flat modules. Section 11.1: dimension of vector space well defined.
27	Dec 11	Section 11.2: rank, nullity, matrix of a linear transformation.
28	Dec 13	Sections 11.3, 11.4: dual space, determinants. Take-home final given out - pdf, tex.