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.
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
Solve each system by substitution: 1. 4x+5y=7 9y=31+2x Solve each system by elimination: 1. 12x-5y=9 3x-8y= -18
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?
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
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).
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?
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
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
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.
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.
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.
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
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
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. 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
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.
Five years ago, you bought a house for $151,000. You had a down payment of $30,000, which meant you took out a loan for $121,000. Your interest rate was $5.75% fixed. You would like to pay more on your loan. You check your bank statement and find the following information. Escrow payment: $211.13 Principle and Interest pay
Explain the difference between an identity, a conditional equation, and an inconsistent equation. Give an example of each and explain why it is so
If Bn is the groups of upper triangular invertible matrices, On is the orthogonal group, SOn is the spacial orthogonal group (n means that the matrices are nxn) and Fp is the prime field, compute the orders of the groups: a) Bn(Fp); b) O2(F7); c) O3(F2); SO3(F3).
The purchasing power P (in dollars) of an annual amount of A dollars after t years of 3% inflation decays according to P=Ae^(-0.03t). How long, rounded to a tenth of year, will it be before a pension of $80,000 per year has a purchasing power of $40,000?
Hello everyone, here's a problem I need help with! Let G be a group, and let K be a subgroup of G. If K_i is normal to G, for each i = 1, 2, ..., n, write K = K_1 (intersect) K_2 (intersect) ... (intersect) K_n. THE QUESTION: If G / K_i is solvable for each i, show that G / K is solvable. I'm sorry if it's hard to r
1. Sachi and Masami have their own secret recipe for making ice cream. They will tell you how much cream they use, but the won't tell you how many marshmallows they put in. Sachi will give you a few clues. She says, "we put in more than 39. We put in fewer than 64. Masami always puts the marshmallows in 3 at a time, with non
The National Association for Women in Science asked recent high school grads if they had taken certain science classes. Of those surveyed, 45 said they had taken physics, 67 said they had taken chemistry and 22 said they had taken both. Seventeen said they had taken neither. How many recent high school grads were surveyed? Show
A bakery makes a special marble rye bread using white flour, light rye flour, and dark rye flower. If they use three fifths as much light rye as dark rye but the amount of white flour is 20% of the amount of dark rye, how much light rye flour is needed to make a 1000 gram loaf? (assume the flour makes up all the weight of the fi
Biology: Some types of worms have a remarkable ability to live without moisture. The following table from one study shows the number of worms W surviving after x days without moisture. x (days) 0 20 40 80 120 160 W (worms) 50 48 45 36 20 3 (
14 Given the following weekly demand figures, what is the MAD at the end of week 5? Week Demand Forecast 1 100 120 2 120 130 3 110 120 4 130 125 5 160 145 A. 4.0 B. 11.8 C. 12.0 D. 13.0 E. 15.0 22. You are given that the forecast for period 6 was 70 while the actual demand for
In a multiple-choice test, each question has five options. Students will get 4 points for each correct answer; lose 2 points for each incorrect answer; and receive no points for unanswered questions. A student does not know the correct answer for one question. Is it to her advantage or disadvantage to guess an answer? Show your