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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.]
Suppose the Phillips curve is represented by the following: πt = πte + 0.20 - 2ut where πte = θπt=1. Assume θ = 0.8, πt=1 = 0.10 (10%) and ut = 0.05 (5%). Calculate πt.
Solve the system of equations using the addition (elimination) method.
Simplify the algebraic expression.
Solve the equation by graphing.
What was the most fruitful cultural interchange in human history and do we have more freedom in the modern world than we did in previous eras?
A production line delivers 9,400 widgets in a year. The widgets are delivered 6 per cart. Each cart takes 5 minutes and 28 secs.
Apply the vertex and intercepts to sketch the graph.
Find the domain and range of the function f(x) = √(5 - x). On a set of axes, give a sketch graph of f(x). Describe, in words, how the graph of f(x) = (x + 1)2 + 4(x + 1) - 1 would look on the same set of axes.
Apply the echelon method to solve the system of three equations in three unknowns.
If you were to let A be a 6 x 14 matrix where the dimension of the row space is 3 (dim(R(A) = 3), what would the dimension of the null space of matrix A (dim(N(A)) be and what would the dimension of the null space of A^T (dim(N(A^T)) be?
solve the system of equations using elimination-x16y-4
Is it possible for an extremely large prime to be expressed as a large integer raised to a very large power? Explain. Are there infinitely many natural numbers that are not prime? If so, prove it.
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