Other stuff:

- Problem Set 1 Due Wednesday January 25 Solutions
- Problem Set 2 Due Wednesday February 1 Solutions,.pdf Solutions, .ps format
- Problem Set 3, .pdf format .ps format Due Wednesday February 8 Solutions,.pdf Solutions, .ps format
- Problem set 4 , .ps format ; Due February 15 Solutions,.pdf Solutions, .ps format ; picture for Problem 4
- Problem Set 5, .pdf format .ps format Due Monday February 27, 8pm Solutions.pdf Solutions.ps
- Problem Set 6; .ps format Due March 15 Solutions.pdf ; Solutions.ps
- Problem Set 7; .ps format Due Wednesday March 22. Solutions.pdf ; solutions.ps
- Problem Set 8 ; .ps format Due Friday March 31. Solutions.pdf ; Solutions.ps
- Problem Set 9 ; .ps format Due Wednesday April 12. solutions.pdf ; solutions.ps

- Lecture 1. Jan. 11.
H 1.1 - 1.3, 2.1. The
integers: well-ordering axiom, division algorithm, the greatest
common divisor, Euclidean algorithm, the greatest common divisor of a
and b as a linear combination of a,b; congruence and congruence
classes.

Suggested Problems: 1.1: 2,6,7,9. 1.2: 1,3--11,14--17,24,26,32,33. 1.3: 1--4, 6,7,10, 11a, 13--16,20--23,27. 2.1:1-7, 8, 10, 16.

- Lecture 2. Jan. 18.
H 2.2, 2.3,3.1,13.1.
Congruence arithmetic; the definitions of a ring, and commutative ring,
a field, an integral domain; examples: the rings Z_n, the complex
numbers as a field of ordered pairs (x,y), the matrix rings,
the finite fields Z_p (when p is a prime). Solving congruences and the
Chinese remainder theorem.

Suggested Problems: 2.2: 6,7,8,9,10. 2.3: 6,8,12. 13.1: 1,8,14. 3.1:1,2,11,14,15,18(!),19,20,21,22,24,30,34.

- Lecture 3. Jan 25.
H 3.1, 3.2.
Basic properties of rings; The notion of units; examples; properties of
integral domains, the notion of characteristic of a ring; division
rings (the example of the ring of real quaternions.

Suggested Problems: 3.1: 13, 16, 24, 29, 35(!).

3.2: 1,6,7,11,14,16,19,22,28,32,37,38. - Lecture 4. Feb. 1
H 3.3, 6.1, 6.2, some 13.3.
Ring isomorphisms and homomorphisms; Ideals; Quotient rings; Z_mn is
isomorphic to Z_m*Z_n when m,n are relatively prime.

Suggested Problems: 3.3: 1,6,7,9,10,17,21,32,39,40. 6.1: 1,2,3,4,7,10,11,13,14,15,16,17,18,21,22,27,35,39,44. - Lecture 5. Feb. 8 H 6.2, 6.3, 4.1 The first isomorphism theorem; an example of a quotient ring: Z[i]/(2-i); classification of homomorphic images if the ring Z; maximal and prime ideals; how to tell whether an ideal is prime/maximal by looking at the correspondiong quotient ring; introduction to polynomial rings.
- Lecture 6. Feb. 15
H 4.1 -- 4.4, 5.1 -- 5.3
Polynomial rings: the definition, notion of degree, many properties
analogous to Z: division algorithm, notion of the greatest common
divisor, F[x] is a PID is F is a field. Irreducible polynomials; roots;
the structure of the quotient
F[x]/(p(x)).

Suggested Problems for the last two lectures: 6.2: 23*,26*,32*,33* (these problems with a * are slightly harder problems, in case you like a challenge)

6.3: 1-4,6,11,20*, 24 4.1: 2-9, 11,12,15-20. 4.2: 3,4,5,7 4.3: 1,3,5,6,9,10,11,13,14,21,23,26.

4.4: 1,3 (just do a couple),5,7(!),8,15,17,19,24(!),25,26 (these two are part of Problem Set 5), 28.

5.1: 3,5,13. 5.2: 5-8; 14a. 5.3: 1,5, 7*,10*. - March 1: Term test
- Lecture 7. March 8
4.5, 4.6, 10.1, 10.2
Irreducibility tests for polynomials, Eisenstein's Criterion, Gauss'
Lemma.
Field extensions.

4.5: 1-6; 8,12-14, 17,20* 4.6: 2,3, 6-8; 10.1: Do as many as needed to get comfortable with vector spaces. - Lecture 8. March 15. 10.1, 10.2, some 10.3 Field extensions; the extensions of the form F(u) where u is a root of an irreducible polynomial over F (they are called "simple extensions"); algebraic extensions.
- Lecture 9. March 22.
10.3, Chapter 15.
Algebraic extensions; Geometric constructions.

Suggested Problems. 10.2: 4,6,9,11,17,21.

10.3: 1-3,6,11, 13,17

15: 3,7,10,16(!) (note that I more or less discussed most of the other problems in class -- e.g. 17,19,21) - Lecture 10. March 29. 10.4 Splitting fields of polynomials.
- Lecture 11. April 5. Automorphisms of fields, especially of the splitting fields of polynomials. The Galois group. How to tell whether a splitting field of a cubic polynomial has degree 3 or 6 over Q. Definition of the Galois group can be found in Chapter 11.1 in Hungerford; however, most of the content of the lecture is not. Please talk to me if you need notes (maybe I'll post them next week). Also, (not included in the exam, etc. -- but just for your interest: you can print two pages on how to solve cubic equations: page 1 , page 2.
- Lecture 12. April 12.
Applications of Galois theory: insolvability of the quintic,
constructibility of regular polygons.

Here is a maple script that illustrates the construction of regular pentagon . To see what it does, download it, start Maple (if you have it), then open the file pentagon.mws inside maple. Here's the end result:

Here is the same file in HTML (sorry, colours are terrible).