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Proof of two Legendre Symbol identities

1) Use that (Z/pZ)* is cyclic to directly prove that (-3/p) = 1 when p = 1 (mod 3). 2) If p = 1 (mod 5), directly prove that (5/p) = 1. I know that with exercise 1, we show that there is an element c in (Z/pZ)* of order 3, and that (2c+1) ^2 = -3. Similarly for exercise 2, there is a c in (Z/pZ)* of order 5, and [(c + c

Aerobics Instructor: Water Consumption Problem

An aerobics instructor says she drinks one cup (8 ounces) of water before she starts her classes and then she drinks 10 ounces every 10 minutes. a. How much water does she drink when she teaches 2 one-hour classes in a row? b. Let W represent ounces of water and N the number of 10-minute intervals the aerobics instructor s

Solving Quadratic Equations With Factoring

Solving Quadratic Equations by Factoring is Chapter 13.7 page 955 1) Quadratic Equation is in STANDARD FORM (ax^2 + bx + c = 0 where a is positive) Factor trinomial: x^2 + bx +c =(X + m)( X +n)=0 ,which :(mn=C),and (m+n=b) (X + m)=0 ---->X = - m , and (X + n)=0----> X= - n 2) Pythagorean Theorem in Right triangle (

Maximal and Principal Ideals

Let delta=sqrt(-3) and R=Z[delta]. This is not the ring of integers in the imaginary quadratic number field Q[delta]. Let A be the ideal (2,1+delta). a) Show that A is a maximal ideal and identify the quotient ring R/A. b) Show that A contains the principal ideal (2) but that A does not divide (2).

Groups and Representations

Please help answer the following algebra questions. Provide step-by-step solutions. (i) Now let G be a group with the presentation G=<a,b|a^7=e,b^3=e,b^(-1) ab=a^2> You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation p:G --> GL(3,C) with p(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(

Forming a linear function from a story

The relation between the cost of a certain gem and its weight is linear. In looking at two gems, we find that one of the gems weighs 0.2 carats and costs $3,548, while the other gem weights 0.5 carats and costs $4,374. Find C(x), the linear function that relates the price of gem to its weight x, treating weight as the independen

Cyclic Modules and Generators

For cyclic Z-modules Zm and Zn with generators a and b, respectively, show that Zm ⊗Z Zn is isomorphic to Z(m,n) with generator a ⊗ b, where(m,n) is the greatest common divisor of m and n.

Prove the following Proposition

Prove the following Proposition: On Z_n, both * and + are commutative and associative. The identity for Z_n with * is [1] and the identity for + is [0]. Please submit response as either a PDF or MS Word file. Infinite thanks. See attachment.

vector question

Under what condition is there a solution to the simultaneous vector equations alpha* x+ beta*y = a and x ^ y = b, for the vectors x and y in terms of given non-zero scalars alpha and beta and given non-zero vectors a and b? Find the general solution to these equations when this condition is satisfied. n.b[ "^" represents t

Working with functions

Let y = f(x) describe the speed y of an automobile after x minutes if the cruise control is set at 60 miles per hour. (a) Represent f symbolically and graphically over a 15-minute period for 0 < x > 15. (b) Construct a table of f for x = 0, 1, 2, . . . ., 6. (c ) What type of function is f?

Proving that a Function is Irreducible

Let p be a prime congruent to -1 mod 4. Show that X^2 + 1 is irreducible in Z_p[X], and hence K = Z_p[X] / (X^2 + 1) is the field of order p^2. Note that K has a multiplication similar to that of the complex numbers.

Nonlinear System

Give all solutions of each nonlinear system of equations, including those with nonreal complex components: 1. X^2+y=2 X-y=0 2. X^2+y^2=10 2X^2-y^2=17 3. -5xy+2=0 X-15y=5

Investment Time and Interest Rate

1. Find t to the nearest hundredth if $1786 becomes $2063 at 2.6%, with interest compounded monthly. 2. At what interest rate, to the nearest hundredth of a percent, will $16,000 grow to $20,000 if invested for 5.25 yr and interest is compounded quarterly?

Inverse Function

a. Write an equation for the inverse function in the form of y=f^-1(x), b. graph f and f^-1 on the axes, and c. give the domain and the range of f and f^-1. If the function is not one-to-one, say so. 1. y=4x-5 2. y=4/x

Parametric equations for a Particle Path

Find the parametric equations for the path of a particle that moves along the circle x^2 + (y-1)^2 = 4 as follows: (a) Once around clockwise, starting at (2,1); (b) Three times around counterclockwise, starting at (2,1); (c) Halfway around counterclockwise, starting at (0,3).

Intersection of surfaces

How do you find the equation of the curve of intersection of the surfaces z = x^2 and x^2 + y^2 =1? How can I show that the curve with parametric equations x = sin t, y = cos t, z = sin^2 t is the curve of intersection of these two surfaces?

Example of a sequence {an} satisfying all of the following

Please show all work. Please see the attachment for the full problems. Problem 1 : Give an example of a sequence {an} satisfying all of the following: {an} is monotonic 0 < an < 1 for all n and no two terms are equal = Problem 2: Let k > 0 be a constant and consider the important sequence {kn}. It?s behaviou


A company makes tops for the boxes of pickup trucks. The total revenue R in dollars from selling the tops for p dollars each is given by where R=p (200-p), where p< 200 (a) Find R when P=$100 (b) Find p when R=$7500 (c) If you have a calculator, use it to solve part (b) numerically with a table of values. Do your answers ag

Riemann Left and Right Hand Sums

Give the following table of values determine by the empirical (ie: the Riemann left and right hand sums) method the area under the function between X=0 and X=70. (hint: use n=7) X 0 10 20 30 40 50 60 70 Y 700 400 300 400 700 1200 1900 2800 A. Graph the function over these x values and explain in your own words what the r

Simple groups

Let G be a group of order n which acts nontrivially on a set of order k. If n>k!, show that G contains a proper normal subgroup. Using the previous result show that if G contains k Sylow p-subgroups, with |G|>k!, then G is not simple.

subsets operations

Let S be a finite set on which a group G operates transitively, and let U be a subset of S. Prove that the subsets gU cover S evenly, that is, that every element of S is in the same number of sets gU.

Class equation

The class equation of a group G is 1+4+5+5+5. a) Does G have a subgroup of order 5? If it does, is it a normal subgroup? b) Does G have a subgroup of order 4? If it does, is it a normal subgroup? c) Determine the possible class equations of nonabelian groups of order 8 and of order 21.

Class equation

a) A group G of order 12 contains a conjugacy class of order 4. Show that the center of G is trivial. b) A group of order 21 contains a conjugacy class C(x) of order 3. What is the order of x in the group?


I have attached six problems (the odd ones from my textbook) that I need assistance with solving. I have no idea how to approach these problems. My professor has assigned the even for homework. I thought if I could see how the odd were worked, I would be able to follow the examples to do the even. Thank you for your assi

Word problems

1. Gennie has a large collection of Barbie dolls, worth a total of $55,000. Within the collection, she has three groups of dolls: Model Collection dolls, International Beauty dolls, and Princess dolls. She determines that the total worth of the Model Collection dolls is $5000 more than the total worth of the International Bea

Fermat's little theorem

How many different substitution ciphers are there? Explain. Fermat's little theorem, which says that if p is a prime number then (n^p) â?' n is always divisible by p is fundamental to many modern methods of cryptography. The important point is that if one restricts attention to possible remainders when divided by p (that

Green House Gases

Green House Gases. Carbon Dioxide CO2 is a green house gas in the atmosphere that may raise average temperatures on Earth. The burning of fossil fuels could be responsible for the increased levels in carbon dioxide. If current trends continue, future concentrations of atmospheric carbon dioxide in parts per million (ppm) could r

Assessing a family of sets

Suppose F is a family of sets. Prove that there is a unique Set A that has the following two properties. a) F is a subset of P(A) and b) For all of B (F is a subset of P(B) then A is a subset of B Should use an upset down A for all of B. For the word then an arrow should be used.