Monday, November 5, 2012
M7210 Lectures 32
Factorization in Commutative Rings III
We assume all rings are commutative and have multiplicative identity (dierent from 0).
Content of a polynomial
Let F be the eld of fractions of A, where A is a UFD. Every el
August 24, 2012
M7210 Lecture 3
Permutations
Let X be a set with n elements, which we label 1, 2, . . . , n.
Denition. A permutation of X is a bijective function : X X .
Table notation. We may describe a permutation by a table of the following form:
1
2
n
August 22, 2012
M7210 Lecture 2. Polynomials
Let F be a eld (e.g., Q, R of C). An polynomial in the variable x with coecients from F
is an expression of the form A = a0 + a1 x + a2 x2 + + an xn , written as a sum of elements
ai F times powers of x of incr
What is a polynomial? Motivating the university-level denition.
The Wikipedia answer is . . .
A polynomial is an expression of nite length constructed from variables and constants using
only the operations of addition, subtraction, multiplication and non-
Monday August 27, 2012
M7210 Lecture 4
Vector Spaces: Denition and Examples
I will assume familiarity with matrix multiplication, with row reduction and row-echelon
form and with the manner in which a system of linear equations may be represented by
an au
M7210 Lecture 5. Vector Spaces Part 2, Matrices
Wednesday August 29, 2012
[delayed due to Hurricane Isaac]
In the present section, we look at how matrices are related to linear maps between vector
spaces of the form Fn and with vector spaces that arise in
Friday September 7, 2012
M7210 Lecture 8. Vector Spaces Part 5
Double dual
If V is nite-dimensional, then V and V have the same dimension, so there is a linear bijection
between them. In fact, there are many, and there is no way (if no basis is specied) o
M7210 Lecture 7. Vector Spaces Part 4, Dual Spaces
Wednesday September 5, 2012
Assume V is an n-dimensional vector space over a eld F.
Denition. The dual space of V , denoted V is the set of all linear maps from V to F.
Comment. F is an ambiguous object.
M7210 Lecture 6. Vector Spaces Part 3, Linear Maps
Friday August 31, 2012
In the previous section, we saw how to associate with an m n matrix with entries from F a
function from Fn to Fm . In the present section, we introduce the general idea of a linear
M7210 Lecture 9. UMPs for Products and Sums
Monday September 10, 2012
Products
Let A and B be sets. The product of A and B is the set
A B := cfw_ (a, b) | a A, b B .
The product comes equipped with projection functions A : A B A; (a, b) a and B : A B
B ;
Wednesday September 12, 2012
M7210 Lecture 10.
Multilinear Functionals
Let V be a vector space of dimension n over F. We will use bold letters v to stand for k-tuples of elements of V ,
i.e., elements of V k .1
Denition. A k-multilinear functional on V is
Wednesday September 19, 2012
M7210 Lecture 12.
Mondays class was cancelled due to a bomb scare that closed down the whole university. This reminds me
of a joke: A bull swallowed a bomb. What do you call it? (Abominable.) The bomb exploded. What do you
cal
M7210 Lecture 11.
Friday September 14, 2012
Determinants (cont.)
Let us adjust notation to match the text. If A is an n n matrix, the entry in its ith row and j th
column will be written Aij . Previously, I called this aij . So, all we are doing is using
Wednesday October 3, 2012
M7210 Lecture 18
Today, we have three goals.
1. Apply the theory of group actions to conjugation and derive the class equation.
2. Apply the class equation to p-groups. We will use the Isomorphism Theorem, which
we have not yet p
M7210 Project
Wednesday September 19, 2012
In this project, you will reproduce the proof of the theorem stated below, which is from
Jordan, P. & von Neumann, J., On Inner Products in Linear, Metric Spaces, Annals
of Mathematics , 36 (1935), 719723.
This
Monday October 1, 2012
M7210 Lecture 17
Group actions.
We ended the last lecture by introducing the set of cosets G/H := cfw_ gh | g G of and arbitrary subgroup
H of a group G. When H is normal, G/H has the structure of a group. The main theme of this le
Friday September 28, 2012
M7210 Lecture 16
Homomorphisms and their kernels
Denition. Let G and H be groups. A function : G H is called a homomorphism if it
preserves the group structure, in the sense that:
i) (eG ) = eH ,
1
ii) for all g G, (g 1 ) = (g )
M7210 Lecture 14
Monday September 24, 2012
Adjoints and the Spectral Theorem
I will be describing the highlights of section III.2 of Knapp. As previously, we assume
V is a nite-dimensional inner-product space over F = R or F = C.
If L : V V is a linear ma
Wednesday September 26, 2012
M7210 Lecture 15
Groups
Denition. A group G is a set equipped with the following data:
a) a designated element e (called the identity element ),
b) a function g g 1 : G G (called inversion ),
c) a function (g, h) gh : G G G (c
Friday October 5, 2012
M7210 Lecture 19
Today, well begin by proving the criterion for recognizing products that we didnt get to last time.
Then, well dene and study semidirect products.
Direct Products.
If K and L are groups, then K L, with operation (k1
Monday, November 26, 2012
M7210 Lecture 38
Sylow Theorems
Review. Suppose X is a G-set, and x X . Recall:
Gx := cfw_ g G | gx = x is called the isotropy group of x, or stabilizer of x.
Gx := cfw_ gx | g G is called the orbit of x.
Counting Formula. Fo
Monday October 15, 2012
M7210 Lecture 23
Abelian Groups III
Today, we are going to set up the machinery we will use to prove:
Theorem. If S is subgroup of Zn , then we can choose a new basis cfw_b1 , . . . , bn of Zn and a new
generating set cfw_t1 , . .
Monday October 17, 2012
M7210 Lecture 24
Abelian Groups IV
We left o last time having nished preparations for the proof of:
Theorem. If S is subgroup of Zn , then we can choose a new basis cfw_b1 , . . . , bn of Zn and a new
generating set cfw_t1 , . . .
Friday, November 9, 2012
M7210 Lecture 34
Cyclic F[x]-modules (continued)
From now on, rather than using coset notation for elements of F[x]/(g ), we will simply use
u0 as a name for 1 + (g ). We let u := xu0 = x + (g ) and ui := xi u0 . In general if f F
Wednesday October 31, 2012
M7210 Lecture 30
Factorization in Commutative Rings
We assume all rings are commutative and have multiplicative identity (dierent from 0).
This lecture is about factorizationi.e., about the multiplicative structure of a ring. Ri
Wednesday, November 7, 2012
M7210 Lecture 33
Cyclic Modules
Let A be a commutative ring with identity and let M be an A-module. Let m M . The
submodule of M generated by m is Am = cfw_ am | a A . (Check this!) We say that M is cyclic
if M = Am for some m
Monday October 22, 2012
M7210 Lecture 26
Commutative Rings I
Reminder:
Denition. A ring is an abelian group A (with operations +, and 0) equipped with a
multiplication A A A; (a, b) a b that satises the following axioms:
a) The multiplication is associati
M7210 Lecture 28
Friday October 26, 2012
Commutative Rings III
We assume all rings are commutative and have multiplicative identity (dierent from 0).
Units and zero-divisors
Denition. Let A be a commutative ring and let a A.
i) a is called a unit if a = 0
Friday November 2, 2012
M7210 Lectures 31
Factorization in Commutative Rings II
We assume all rings are commutative and have multiplicative identity (dierent from 0).
Greatest common divisors in UFDs
Let A be a UFD. In A, very non-zero non-unit is a nite
Friday October 12, 2012
M7210 Lecture 22
Abelian Groups II
Sums of R-modules
If M , is any set of R-modules, then M denotes the subset of the cartesian product M
consisting of those elements that are non-zero for at most nitely many indices of . (If is ni