Let α ∈ C be a root of x^2 + x + 1 ∈ Q[x]. For...

Let α ∈ C be a root of x^2 + x + 1 ∈ Q[x]. For γ = 3 + 2α ∈ Q(α), find γ^ −1 as an element of Q(α).

Let a = 3 + 2(2^(1/3)) + 4^(1/3) and b = 1 + 5(4)^(1/3) belong to Q( 2^(1/3)). Calculate a · b and a −1 as elements of Q( 2^(1/3)).

Expert Solution

Colby Messinger answered 2 years ago

Let α be a root of x^2 +1 ∈ Z3[x]. Show that Z3(α) = {a+bα :...

Let α be a root of x^2 +1 ∈ Z3[x]. Show that Z3(α) = {a+bα : a, b ∈ Z3} is isomorphic to Z3[x]/(x^2 + 1).

let α ∈ C be a zero of the polynomial t^3 − 4t + 2 =...

let α ∈ C be a zero of the polynomial t^3 − 4t + 2 = 0 and let R = {a1 + bα + cα^2 : a,b,c ∈ Z}. Show that R is a integral domain and Show that α − 1 and 2α − 1 are units in R. [Hint: what if x = t + 1?

Differentiate. 1a) p(x) = 4th root of (2x-3/x^3) b) q(x)= (2 sinx)/(1-cosx) c)r(x)= sin(csc^3(x^4)) d) U(x)=...

Differentiate. 1a) p(x) = 4th root of (2x-3/x^3) b) q(x)= (2 sinx)/(1-cosx) c)r(x)= sin(csc^3(x^4)) d) U(x)= ((cube root of x^2) sinx -2x-3)/sq. rt. x

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α ≠ Q 2) if r ∈ α and s ∈ Q satisfies s < r, then s ∈ α. 3) if r ∈ α, then there exists s ∈ Q with s > r and s ∈ α. Let α + β = {r + s | r ∈ α and s ∈ β}. Show that the set of all cuts R with the...

Prove (for example, by calculating each side) (a) Q[ √ 2, cube root(2)] : Q =...

Prove (for example, by calculating each side) (a) Q[ √ 2, cube root(2)] : Q = [Q[ √ 2] : Q] · [Q[ cube root(2)] : Q] (b) Q[ √ 2, fourth root (2)] : Q not equal to [Q[ √ 2] : Q] · [Q[ fourth root (2)] : Q]

Write a Matlab function for: 1. Root Finding: Calculate the root of the equation f(x)=x^3 −5x^2...

Write a Matlab function for: 1. Root Finding: Calculate the root of the equation f(x)=x^3 −5x^2 +3x−7 Calculate the accuracy of the solution to 1 × 10−10. Find the number of iterations required to achieve this accuracy. Compute the root of the equation with the bisection method. Your program should output the following lines: • Bisection Method: Method converged to root X after Y iterations with a relative error of Z.

Q. Let A, B independent events, with P(A) = 1/2 and P(B) = 2/3. Now C...

Using Baynesian estimation. 1. Let X is Poi(Ꝋ). Let Ꝋ be Γ(α, β). Show that the...

Using Baynesian estimation. 1. Let X is Poi(Ꝋ). Let Ꝋ be Γ(α, β). Show that the marginal pmf of X (the compound distribution) is k1(x) = (Γ (α + x) β^x) / (Γ(α) x! (1 + β)^(α+x ); x = 0, 1, 2, 3, …, which is a generalization of the negative binomial distribution.

Let X = {1, 2, 3, 4}, Y = {a, b, c}. (1) Give an example...

Let X = {1, 2, 3, 4}, Y = {a, b, c}. (1) Give an example for f : X → Y so that ∀y ∈ Y, ∃x ∈ X, f(x) = y. 1 2 (2) Give an example for f : X → Y so that ∃y ∈ Y, ∀x ∈ X, f(x) = y. (3) Give an example for f : X → Y and g : Y → X so that f ◦ g = IY

. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all...

. Suppose that the sequence (xn) satisfies |xn –α| ≤ c | xn-1- α|2 for all n. Show by induction that c | xn- α| ≤ c | x0 - α|2n , and give some condition That is sufficient for the convergence of (xn) to α. Use part a) to estimate the number of iterations needed to reach accuracy |xn –α| < 10-12 in case c = 10 and |x0 –α |= 0.09.

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