
Renzo's class 


MWF 10 – 10:50 am 


ENGRG E 103
// in the oval with good weather 







TEXTBOOK:
Abstract Algebra: Theory and Applications
by Thomas W Judson
Publisher:
available freely from:
http://abstract.ups.edu/index.
MOTIVATIONAL STUFF
If you want to
know why you should study abstract algebra, I think the best answer is: to train your brain to think! Just how when you do pushups at the
gym, your goal is to get stronger, not just to perform *the perfect pushup*.
However, if you stick with abstract math long enough (a lot longer than this
one semester course) – then you will find that there are powerful
applications of abstract algebra. Below is a small compilation of a few of
them, if you are curious!
STUFF
Exercises in preparation
for baby exam (pdf) (tex file here if you want)
Exercises in
preparation for exam 2 (pdf) (tex file here if
you want)
EXAMS
LaTeX CHEATSHEET:
Office
hours : the official office
hours for this class are right after class. However, if these times are not
convenient for you, you are very welcome to make an appointment and come ask
questions, make comments, or just chat. You can also try showing up at my door
anytime. But I might tell you to come back at another time if I am immersed
into something else.
The tables of the law for this class are contained in the
following document:
Syllabus
Homework: math is not a sport for bystanders. Getting
your hands dirty is important to make sure that things sink in and you
are not just spending a semester assisting to my creative rambling. Homework
will be due pretty much every Friday (but see below for the uptodate
information). I will NOT grade all the problems I assign. If one of the
problems you feel unsure about doesn’t get graded, please don’t go
“Whew! Lucky one!” but rather come ask me about it. The point is
you understanding, not pretending to!!
Important: the number of exercises and pages
are mismatched between the electronic and paper versions of the book. I will
always refer to the electronic version available following the link above
(latest edition), since not everyone owns the paper version of the book!
DATE DUE: 

Jan 27 
Exercises 1, 2, 7, 9, 18, 19 pages 1820. Additional exercise: prove that, given two sets A,B A is a subset of B if and only if the intersection of A with the complement of B is empty. Write
a paragraph to explain how this fact relates to proving a theorem of the form 
Feb 3 
Exercises 22, 26 pages 1820 are recommended as preparation for the exam, together with the worksheet. Baby Exam 
Feb 10 
1) Write down in full
detail and paying great care to
your exposition two different proofs for the identity: 1+2+3+…+n= ½
n(n+1) 2) Compute the sum of the
first n squares in two different ways: First, make a wild guess
that the answer is a degree 3 polynomial in n 1+4+9+…+n^2=
An^3+Bn^2+Cn +D Plug in values for n to
determine A,B,C,D by solving a linear system of 4 equations in 4 variables.
Then prove the formula by induction. In the second case, use
the identity (n+1)^3= n^3+3n^2+3n+1 to generalize to this
case one of the proofs seen in
class for the sum of the first n
integers. 
Feb 17 
Exercises 8, 12 page 3233 1) Prove by induction the following statement: the sum of all internal angles of a convex polygon with nsides is
(n2)180 degrees. (note: in this case the statement makes sense only
for n greater than or equal to 3, and therefore your base case should be
n=3). 2) Let X be a set with n elements. Given three integers a,b,c such that a+b+c=n, how many ways can you subdivide X into three disjoint subsets with a,b and c elements? (I am looking for an answer in terms of products
and quotients of factorials, just like in case of the binomial coefficient). 3) Challenge: can you generalize problem 2) the following way: X a set
with n elements. a_1, a_2, …,
a_k k integers adding up to n. How many ways can you subdivide X
into k subsets S_i such that the cardinality of S_i is a_i. This number is
called the multinomial coefficient n choose a_1, …, a_k. As an extra challenge: can you figure
out how (If you tackle the extra challenge. First make a
guess to what the formula should be. Verify it in for some small values of n
and k. Then try to prove it either by induction or just by explaining why the
coefficients are what they are). 
Feb 24 
Exercises 18, 23, 25, 31 page 34 
Mar 2 
Answer Questions 1,2,3,4,5 on the handout “Rings and Fields” 
Mar 9 
Study for the midterm! Midterm today! 
Mar 23 
1)If you are born in month “x < 12”, do problem x on the handout “Rings and Fields”. If you are born in December, then do the challenge (section 4.4) 2) If you are born in day x, then do problem [x] mod 11. If that turns out to be the same number as in 1), then pick a second problem of choice. 
Mar 30 
Exercises 7, 10, 24, 29, 30, 38, 44, 45, 47, pages 5355. 
Apr 6 
On the online version of the book exercises: 1(c),(d) [to make this question nontautological, you need to insert the adjective nontrivial between “every” and “subgroup”],(e), 2 – (a),(b), 11, 14, 23, 29, 30. Pages 7174. (1) Give a formula for the order of a general element [m] in Z/nZ. Proceed by first experimenting with small values of n until you understand what is going on. Then write a general proof. (2) Complete and prove the following statement. There exists a group homomorphism f: Z/nZ à G sending [1]à g if and only if the order of g…(fill in). If the order of g …(fill in), then f is an isomorphism between Z/nZ and <g>. 
Apr 13 
On the online version of the book exercises: 1, 2, 8, 9, 18, 23, 27, 31, 33, 35 pages 9093. 
Apr 27 
Exercises 1(a),(b),(e), 5, 6, 14 (a),(c),(d) pages 163164. 

