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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.
A chemist is mixing a solution containing 20% bromine with a second solution containing 35% bromine. He wishes to prepare 200 oz.
Minfly Golf Company manufactures golf balls at Philadelphia, Chicago, and Denver. Cases of golf balls are shipped to warehouses in Orlando, Dallas.
Jim has 400 ft of fence material and needs to enclose an area of 10,000 square feet for his garden as the diagram depicts.
Let's explore how consecutive integer problems work. We need to find two consecutive page numbers in a book such that the sum of the page numbers is 11.
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.
On Z_n, both ? and ? are commutative and associative. The identity for Z_n with ? is [1] and the identity for ?is [0].
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).
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.
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.
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.
In solving the equation (x + 1)(x - 2) = 4, Eric stated that the solution would be x + 1 = 4 => x = 3 or (x - 2) = 4 => x = 6.
Solve each formula for the indicated letter. Assume that all variables represent nonnegative numbers. A = pr^2 + prs, for r
Identify the key components of the graph (minimum or maximum, vertex, roots) and give a brief description of what each represents.
Different pets need to have different areas. For example, a horse needs more space than a dog. Determine an area that is appropriate for your pet.
Let a_n from n=0 to infinity be a sequence of real numbers which converge to 0, i.e. lim n-->infinity a_n=0.
You have probably seen the following trick to sum this series: if we call the above sum S, then if we multiply by 2, we obtain: 2S=2+1+1/2+1/4+...=2+S
Find an arithmetic sequence that gives the amount depreciated each year by each method.
Quadratic equations, which are expressed in the form of ax2 + bx + c = 0, where a does not equal 0, may have how many solutions?
You are choosing between two health clubs. Club A offers membership for a fee of $50 plus a monthly fee of $25.
You are doubling a receipe that calls for 3/4 of a cup of milk. How much milk will you need?
Assume the operator R(theta) transforms a vector v (in two-dimensional real space) by swinging it around counterclockwise through an angle theta.
Begin by solving the linear equation for Y. This will put the equation in slope-intercept form.
The equation of a line is given below. Find the slope of a line that is Parallel to the line with the given equation.