(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1. (1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]
for the sample space {w,x,y,z}, p(x)=0.2, p(y)=0.15, p({w,y})=0.7, p({x,z})=0.3. Find p(w), p(z), and p({w,x,z}), using the properties of probability.
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3. Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
Let X ~ N(0, 1), and let Z ~ Unif{-1, 1} (i.e. P(Z = -1) = P(Z = 1) = 1/2) be independent of X. Let Y = ZX. What is the distribution of Y? Show that X and Y are uncorrelated. Are X and Y independent?
6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z < 0 oriented by the upward- pointing unit normal. 6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z
Let p be a prime. Show that Zp(X)/(X2+1) is a field iff the equation x2=-1 has no solution (mod p).
1. Determine the Fourier transform of r(t) = e-t/2u(t) and sketch (a) X(w) (b) ZX(w) (c) Re{Xw)} (d) Im{Xw)}
Recall the formula for a proportion confidence interval is p^?zp^(1?p^)n?????????<p<p^+zp^(1?p^)n????????? Thus, the margin of error is E=zp^(1?p^)n????????? . NOTE: the margin of error can be recovered after constructing a confidence interval on the calculator using algebra (that is, subtracting p^ from the right endpoint.) In a simple random sample of size 59, taken from a population, 20 of the individuals met a specified criteria. a) What is the margin of error for a 90% confidence interval for p, the population...
8. Let p be a prime number. Define -c0t}cQ ZAp) Prove that Zp) is a subring of Q Prove that Z is a subring of Z Show that the field of fractions of Zp) is isomorphic to Q