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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.]
Two sessions of swimming lessons were held at a pool. In the first session 40 students attended. Of these 40 students 60% were girls. How many girls attended the first session of swimming
Find the equation of the straight line perpendicular to the line and find the equation of the parabola in general from whose vertex.
Suppose that a function f reals to the reals is a uniform limit of a sequence of polynomials on the reals. Show that f is a polynomial.
Which type of correlation is suggested by the scatter plot? Write the equation of a line that is perpendicular to the given line and that passes through the given point.
Find the equation of the tangent line
Use the following graph of a function f to determine x or y to the nearest integer as indicated. Some problems may have more than one answer.
Create a class Polynomial and a corresponding test class that is used to evaluate a polynomial function of x: P(x) = a0 + a1 * x + a2 * x ** 2 + ......+an * x ** n; ai is coefficient i, and x ** n means x to the power of n
Solve the linear equation
Group theory applications.
If the trend continues, determine when the percent of women in the workforce will equal the percent of men. (Provide a graph showing the linear trends and the breakeven point).
Leon drove 270 to the lodge in the same time as Pat drove 330 miles to the lodge. If Pat drove 10mph faster than Leon, how fast did each of them drive?
Solve the given fraction
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