Fall 2007
MATH 4361 – Modern Algebra
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Date |
Section |
Assignment |
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Aug 15 |
The
Well-Ordering Principle and the Principle of Mathematical Induction. |
Read
Section 1.1. Pay special attention to examples. Suggested problems: Section 1.1: 1-6; 10, 11,
12(those of you who took Real Analysis must have seen this problem before). Class
handout: 1, 2, 4. |
|
Aug 20 |
Relations.
Equivalence relations. Classes of equivalence. |
Hw#1: from handout given on Aug. 15:
1(d, f), 2(c). Section 1.2: 7, 8, 11. Read
the pages on Equivalence relations and the examples from the text. Suggested
pb: 7, 9. See some additional
problems. Read
1.2 and 1.3. |
|
Aug 22 |
Arithmetic
and divisibility. Gcd and
the |
Read 1.2 and 1.3. See Hw#1 revised
above. Suggested pb: section 1.3: 1, 2, 3, 4, 5, 6, 7, 8, 9. See some sample
solutions to text problems. |
|
Aug 27 |
Finding
the lcm(a,b); Diophantine equations. |
Hw#1 due Suggested
pb: section 1.3: 10, 11. See also from above. Remember
also the question addressed in class: If gcd(a,b)=1 then does the equation
ax+by=c, (c integer) always have
solutions in Z? Here
are some more problems
for you. |
|
Aug 29 |
Section
1.4. Prime numbers and the Fundamental Theorem of Arithmetic. |
Hw#2:
Task 1(page 15); page 25: 9, 11; page 31: 2, 4. Here
are some notes on problem 5 section
1.4. Read
section 1.4 and go over the result with paper and pencil trying to understand
the proofs. Look at
sections 1.5 and 1.6. Maybe we can have presentations on Fibonacci numbers,
and using |
|
Sept 3 |
Labor
Day. No classes. |
Have great
weekend, but don’t forget Modern Algebra! |
|
Sept 5 |
Theorems
of Euler and Fermat (section 1.7) Selected solutions to problems from
Hw#1 are now posted. |
Hw#2 due Read section 1.7. Try to understand on your own the
proof of Euler’s theorem (Theorem 3). Suggested problems: section 1.7: 1, 2, 3, 4, 6, 7. If
you like a challenge try 9 as well. Selected solved
problems from 1.7 A practice test
is posted on the Resources page. |
|
Sept 10 |
Review
for exam |
|
|
Sept 12 |
Exam 1 |
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|
Sept 17 |
Modular
arithmetic. Congruences. |
Read
Section 2.1. Read ahead on linear congruences and solving them. Suggested
pb.: section 2.1: 1, 2, 3, 5, 6, 7, 8, 15. Try 9, 11, 12. |
|
Sept 19 |
Solving linear congruences. |
Read
section 2.1 again. More pb: 11, 12, 13, 14, 15, 17, 18. Read
2.2 Worksheet 2: Divisibility tests. I will not be spending time in class
with them, but there are some nice and quite elementary results. Hw#3:
section 2.1: 2, 3, 9(ii, iv, v), 11(ii, iii), find in both
mod 9 and mod 15. Section 2.4: 3, 5(ii, v, vi), 12. |
|
Sept 24 |
The ring of congruences mod m.(section 2.4) |
Suggested
problems: section 2.4: 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. See
some solved problems and some suggested ones in the notes. |
|
Sept 26 |
Rings
and Fields (section 2.5) |
Hw#3 due For
next class: read 2.5 Suggested
problems: 2.5: 1, 2, 3, 4, 5, 6, 7, 8, 11, 12… I would better say all of
them, or almost all of them. Enjoy
the weekend! |
|
Oct 1 |
The
Field of Complex Numbers.
|
Read
2.6, “to the teacher” notes included. Also, read 2.7 and 2.8 Suggested
problems: 2.6: 1, 2, 6, 7, 8, 9. Have a
look at the problems at the end of the chapter, on page 108. Look closer at
18. More to
come… |
|
Oct 3 |
More on
Complex numbers. |
Some class notes and problems. |
|
Oct 5 |
Review in CL 2003 from |
|
|
Oct 8 |
Review |
Additional
problems on 2.6:10, 11, 12, 13, 14, 17. Page
109: 3, 4, 5, 10, 11, 12, 16, 18, 19, 20, 22. |
|
Oct 10 |
Exam on
Chapter 2 |
Hw #4:
problems 1, 2 in the class notes above, pb. 22 page 109 (text). Read
ahead Chapter 3: 3.1 and 3.2. They are not difficult. Have a
great weekend! |
|
Oct 15 |
Polynomial arithmetic: Chapter 3 |
Read
3.1. Hw#4 due postponed until Wednesday Added to hw4: section 3.1: 3(ii),5(iii), 10. Suggested
problems: work as many as you can. |
|
Oct 17 |
Check the Employer showcase. It is today right under your nose
(i.e in Science Atrium and Clendenin lobby)! |
Hw#4 due |
|
Oct 22 |
Some problems on factoring polynomials. Derivatives of polynomials: 3.3 |
Read
3.3. Suggested
problems: section 3.2: 3, 4, 5, 6, 7(you may use the method we used in class
or the quadratic formula) Section
3.3: Tasks on page 130: 1, 2, 3. Hw#5:
prove Proposition A page 129, Task 2 on page 130; section 3.2: 4(iii), 5(iii do it
also in Z_3[x]). |
|
Oct 24 |
Factoring in Z[x] and Q[x]. |
Hw#5
continued from above: section 3.4: 5(ii), 7(ii). |
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Oct 26 |
I will be in CL 1005 between 9-11 and in my
office after that if anybody needs me! |
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|
Oct 29 |
The
Fundamental Theorem of Algebra. Polynomial congruences (section 3.7). |
Hw#5 due Read
section 3.5. pages 136-142. You may skip the proof of the theorem. Read
section 3.7: pages 150-154. Suggested
problems: Section 3.5: 1, 2, 4, 6, 7, 8(try it). Page 146: 1, 2, 3, 6. Section
3.7: 1, 2, 3, 4. |
|
Oct 31 |
Polynomial
congruences. |
Suggested
problems: section 3.7: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. |
|
Nov 5 |
Review
for Exam 3 |
Look at
the Resources page for new links to texts and study guides for modern algebra
and some interesting articles. Some extra problems to keep you company
over the weekend! |
|
Nov 7 |
Exam #3: Chapter
3 what was covered in class |
Solutions to selected problems
from the list of practice problems from above. |
|
Nov 12 |
Introduction
to groups.(section 4.1) |
Read
section 4.1. Suggested
problems: 1, 2, 3, 4, 5, 6, 7, 8, 9-19 (they are all interesting). |
|
Nov 14 |
More on
groups (still 4.1) |
Read
carefully the examples in 4.1. Read
sections 4.2 and 4.3. They are very interesting! Hw#6L (the last hw of the semesterJ): ex. 6, 16, 33, 49 class
handout; Here
are some solved problems on
groups. |
|
Nov 19 |
Subgroups.(section
4.4) Cyclic subgroups. |
Hw#6 is due by Tuesday, Nov 20, 3pm. Suggested
problems: All in section 4.4 are great problems. Enjoy them! Some solved problems on section 4.4 |
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Nov 26 |
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Nov 28 |
Exam #4:
Chapter 4, sections 4.1 and 4.4 |
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Dec 3 |
Last day of classesJ, but I am going to
miss youL guys. |
If you
decide you don’t need to take the final I wish you Happy Holidays, have a
wonderful end of semester and year and best of luck in the future! If you are
taking the final, let me know if you want to do a review session this week. Since
not all of you came to me to see the grades I will post them on WebCT. The
final exam is not going to lower the grade, I want to offer a chance to you
to get a better grade or a passing grade even if your average may seem quite
low at this moment. I will
be here this week, but I will try to offer a review on Friday, around 1:30 if
I know there will be more of you coming. If I hear from you I will try to set
the day and time to accommodate more. |
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Dec 7 |
Review in CL 2003 1:30-3Pm |
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