# MAT 401 Polynomial Equations and Fields.

### Text: T. Hungerford,  "Abstract Algebra,  an Introduction" (preferably the second edition).

News:
• The last office hours: Thursday April 20, 1-2pm; Tuesday April 25, 2-4pm.
• Topics to review for the final exam. .ps format
• An example of a splitting field, page 1 ; page 2 . This is an example taken from Chapter 20 in Gallian's book. It describes the splitting field of some polynomial over a finite field, and the decomposition of that polynomial into linear factors over that splitting field. See Example 6 (everything else is there just because it happens to be on the same page).
• Solutions to the midterm .pdf ; .ps format

• Other stuff:
• Course description includes the marking scheme
• Formula for roots of cubic equations, page 1 ; page 2 . This is just for those who are curious how to solve cubic equations.

### Lecture Outlines and Suggested Problems:

#### "H" stands for Hungerford.

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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*.

7. March 1: Term test

8. 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.

9. 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.

10. 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)

11. Lecture 10. March 29.  10.4 Splitting fields of polynomials.

12. 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.

13. 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).