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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.]
Prove the following vector calculus identity in R 3 , where f is a twice continuously dierentiable scalar eld and F is a twice continuously dierentiable vector fi eld
Assume Jim had executed 15 "Splits" before his last split of 20 seconds. If his eventual time in the road race is 4:05 , what was the average time for one of his earlier splits?
Determine the Minimin and Minimax regret
Let R be a set with 2 laws of composition satisfying all ring axioms except the comutative law for addition
Find the vertex.
Provide probability distribution, histogram, and find the expected value of a coin that is tossed 3 times
Escher style tessellation
Finding the solution for the specified variable in the given equation and Solve: 3xy = wyz
Use Mad to compare simple moving & weighted moving average
Find the supply and demand for particular value when supply and demand functions are given and How many videotapes are demanded at a price of $20?
Among the professionals you have interviewed for your article, were several state and federal government spokespersons who use linear equations in a variety of ways.
Find the condition that the curve y = mx + c to be a tangent to the parabola y^2 = 4 ax and also determine the point of contact.
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