Math 536 Homework 4
Due: Friday, February 8
1. Let : B C be a surjective homomorphism of groups. Show that if F is free abelian
and : F C is any homomorphism, then there exists a homomorphism : F B
making the diagram below commute, i.e. such t
Unit 6: Stoichiometry
The Big Ideas
Balance chemical equations to preserve the Law of Conservation of Mass.
Describe the number of atoms that participate in a chemical reaction.
Describe the number of molecules that par
Unit 5: Chemical Reactions
The Big Ideas
Law of Conservation of Mass:
o matter is neither created nor destroyed
o atoms that go into a chemical reaction as reactants, must come out as products
o reaction coefficients are used to balance the number of ato
By watching one of these videos, observe the child and fill out the form.
Preschool Observation Form
I. Description of child: _
Location of video (outside, playground, etc) : _
Video# _ Video #_
Host Evaluation Form
Host Signature must be in ink and cannot be an electronic
signature or it will not be accepted!
Dates/Hours of Service:
Student: Name 5 things you did/le
Math 536 Homework 8
Due: Friday, March 22
1. (a) (Euclids algorithm for nding the gcd.) Let a1 , a2 be nonzero elements of a
Euclidean domain R. Dene ai and qi recursively by a1 = q1 a2 + a3 , ai =
qi ai+1 + ai+2 where (ai+2 ) < (ai+1 ). Show
Math 536 Homework 12
Due: Friday, April 26
1. It is a fact that, if p is prime, then Sp is generated by a transposition and a p-cycle.
(a) Show that if a polynomial f (x) Q[x] is irreducible of prime degree p and has
exactly p 2 real roots, th
Math 536 Homework 10
Due: Friday, April 12
1. Let R be a domain containing a eld k as a subring. Suppose that R is a nite
dimensional vector space over k under the ring multiplication. Show that R is a eld.
2. Let F be a eld of characteristic
Math 536 Homework 6
Due: Friday, March 1
1. Let G be a simple group of order 60. Show that G A5 . (Hint: Consider the dierent
possibilities for the number of Sylow 2-subgroups.)
2. Let p, q be distinct primes. Prove that any group of order p
Math 536 Homework 1
Due: Friday, January 18
1. Let G be a group, and let H1 , . . . , Hk be subgroups of G. We say that G is an internal
direct product of the subgroups Hi if the map
(h1 , . . . , hk ) h1 h2 hk : H1 H2 Hk G
is an isomorphism o
Math 536 Homework 9
Due: Wednesday, March 27
1. Suppose that R is a commutative ring with identity such that every submodule of every
free R-module is free. Show that R is a PID.
2. Let R := Z[X ]. Give an example of a nitely generated R-modul
Math 536 Homework 11
Due: Friday, April 19
1. Give explicit generators for the subelds of C which are splitting elds of the following
polynomials over Q, and nd the degree of each such splitting eld.
(a) X 3 2
(b) (X 3 2)(X 2 2)
(c) X 2 + X +
Math 536 Homework 7
Due: Friday, March 15
1. Let D be an integer 1 and let R be the set of all elements a + b D with a, b Z.
(a) Show that R is a ring.
(b) Let N : R Z be the norm map, i.e. the map given by
N (a + b D) = (a + b D)(a b D).
Math 536 Homework 5
Due: Friday, February 22
1. Let G and H be nite groups of relatively prime order. Show that Aut(G H ) is
isomorphic to Aut(G) Aut(H ).
2. For which primes p and positive integers n is every p-Sylow subgroup of the symmetric
Math 536 Homework 2
Due: Friday, January 25
1. Show that a group of order 2m, m odd, contains a subgroup of index 2. (Hint: You
may use Cayleys theorem, Corollary 4, page 120 in Dummit and Foote, or Jacobson,
Corollary on page 38.)
2. Let K be
Math 536 Homework 3
Due: Friday, February 1
1. Let G be the group of invertible 4 4 matrices over the complex numbers, and let M
be the set of all 4 4 complex matrices.
(a) Consider the action of G G on M given by (g, h) acts on m by the matri
SECTION 9 STUDY GUIDE
Lesson 1 Funding a Business:
1. Types of funding
a. Debt Funding - when you get money by borrowing it and promising to pay it back
EXAMPLES- Short-term and long-term loans, credit cards, and lines