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1.
a. show that every subfield of complex numbers contains rational numbers
b. show that the prime field of real numbers is rational numbers
c. show that the prime field of complex numbers is rational numbers
2.
a. Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R.
b. Show that ([2]x + [1])^2 = [1] in (integers modulo 4)[x] Conclude that the statement in part (a) may be false for the commutative rings that are not domains. [ An element z element of R is called a nilpoint if z^m = 0 for some integer m greater than or equal to one. For any commutative ring R, it can be proved that a polynomial f(x) = a(sub 0) + a(sub 1)x +...+a(sub n)x^n element R[x] is a unit in R[x] if and only if a(sub 0) is a unit in R and a(sub i) is nilpoint for all I greater than or equal to 1.]
Graphical Probability and Graphical Cumulative Probability. A fair coin is tossed 10 times. Let X be the number of tails observed.
Calculation of a Laplace Transform from first principles
You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
Suppose that V and W are vector subspaces of Rn. If I define: How can I prove that V+W is also a vector subspace of Rn and ALSO how could verify that (for example)
Create an indicator variable for identical twin.
Odds of Hitting a Bankrupt on Wheel of Fortune. Assume that the Wheel of Fortune has 24 slots, each subtending 15 degrees. There are two bankrupt slots.
Please solve for the following: Find a particular solution to, Solve: A Second Order ODE y'' + 5y' +4y = -13te^(3t)
Find the validity of argument form.
Express the volume of this region as a single triple integral in rectangular coordinates and Express this integral in terms of one or more integrals in the order dz dx dy.
A school had a very unusual tradition involving its 1000 students and its 1000 lockers. On opening day, after the head of the school had closed all the lockers, a student walked by and opened every single one.
Real Life Application of Linear Eqations : A mathematician has developed a model indicating a woman with x years of education can expect to earn y = 1200x + 6300 dollars a year.
Evaluate the matrix of transformation
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