MAT 401 Polynomial Equations and Fields.
Text: T. Hungerford, "Abstract Algebra, an
Introduction" (preferably the second edition).
The last office hours: Thursday April 20, 1-2pm; Tuesday April 25,
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 ;
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.
Problem sets with solutions:
- Problem Set 1 Due Wednesday January 25
- 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
- Problem Set 6; .ps
format Due March 15 Solutions.pdf ; Solutions.ps
- Problem Set 7; .ps
format Due Wednesday March 22. Solutions.pdf
- Problem Set 8 ;
Due Friday March 31. Solutions.pdf ; Solutions.ps
- Problem Set 9 ;
.ps format Due Wednesday April 12.
solutions.pdf ; solutions.ps
Below you will see a brief outline of each lecture with the
list of suggested problems. You're strongly advised to attempt
all the suggested problems and seek help as necessary, right after the
corresponding lecture. The problems that come from the group "A" after
each section in Hungerford are "core problems" -- you should be
comfortable with them to pass the course. So please don't wait until
the last minute to try them and ask questions. Good luck!
Lecture Outlines and Suggested Problems:
"H" stands for Hungerford.
- 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
Suggested Problems: 1.1:
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:
6,8,12. 13.1: 1,8,14.
- 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(!).
- 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:
- Lecture 5. Feb. 8
H 6.2, 6.3, 4.1
The first isomorphism theorem; an example of a quotient ring:
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
- 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
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
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'
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");
- Lecture 9. March 22.
10.3, Chapter 15.
Algebraic extensions; Geometric constructions.
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.
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).