A field F is said to be perfect if every polynomial over F is separable. Equivalently,
every algebraic extension of F is separable. Thus fields of characteristic zero and
finite fields are perfect. Show that if F has prime characteristic p, then F is perfect
if and only if every element of F is the pth power of some element of F. For short we
write F = F p.

prove that a translation is an isometry
"i want the prove by using a parallelogram and proving that the two side are congruent please"

Multivariate analysis

Using the data provided, perform the following analysis:

• Determine the explanatory and response variables.
• Run a multivariate regression analysis on all three variables.

Interpret the results by answering the following questions:

• What is the calculated correlation coefficient? Do the sales figures correlate with the marketing expenditure and price?
• Comment on the coefficient of determination. What percentage of the response data can be explained by the explanatory variables?
• Determine the multiple regression line equation in the form:

sales^ = (intercept) + (coefficient)× marketing + (coefficient)× price

• Using the regression equation formulated, what is the amount of expected sales (in pounds), if the price is set at £3.50 and the amount spent on marketing is £300?
• Interpret the variables in the regression equation. What impact does each of the factors (marketing and price) have on the sales figures?

What is the last digit of 123456789012345678×609 in base 6. Don’t try to compute the product.

30r+45d

10r+15d≤1800

20r+25d≤2500

6r+15d≤1500

s.t.

a) Use the Graphical Solution Method to solve the linear programming problem (label all functions, corner point points, and axis).

r≥50
2d-1r ≥0
r,d≥0 or all variables non-negative

7. Let E be a finite extension of the field F of prime characteristic p. Show that the
extension is separable if and only if E = F(Ep).

let E be a finite extension of a field F of prime characteristic p, and let K = F(Ep)
be the subfield of E obtained from F by adjoining the pth powers of all elements of
E. Show that F(Ep) consists of all finite linear combinations of elements in Ep with
coefficients in F.

Let A = {r belongs to Q : e < r < pi} show that A is closed and bounded but not compact

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)).

The data set cherry.csv, from Hand et al. (1994), contains measurements of diameter (inches), height (feet), and timber volume (cubic feet) for a sample of 31 black cherry trees. Diameter and height of trees are easily measured, but volume is more difficult to measure.

(i) Suppose that these trees are a SRS from a forest of N = 2967 trees and that the sum of the diameters for all trees in the forest is tx = 41835 inches. Use ratio estimation to estimate the total volume for all trees in the forest. Give a 95% CI.

(ii) Estimate the average timber volume and construct the 95% CI.

"Diam","Height","Volume"
8.3,70,10.3
8.6,65,10.3
8.8,63,10.2
10.5,72,16.4
10.7,81,18.8
10.8,83,19.7
11,66,15.6
11,75,18.2
11.1,80,22.6
11.2,75,19.9
11.3,79,24.2
11.4,76,21
11.4,76,21.4
11.7,69,21.3
12,75,19.1
12.9,74,22.2
12.9,85,33.8
13.3,86,27.4
13.7,71,25.7
13.8,64,24.9
14,78,34.5
14.2,80,31.7
14.5,74,36.3
16,72,38.3
16.3,77,42.6
17.3,81,55.4
17.5,82,55.7
17.9,80,58.3
18,80,51.5
18,80,51
20.6,87,77

Let W be a subspace of Rⁿ. Prove that W⊥ is also a subspace of Rⁿ.

Find the characteristic equation and the eigenvalues (and corresponding eigenvectors) of the matrix. 2 −2 5 0 3 −2 0 −1 2 (a) the characteristic equation (b) the eigenvalues (Enter your answers from smallest to largest.) (λ1, λ2, λ3) = the corresponding eigenvectors x1 = x2 = x3 =

Find all the six roots of z = 16−16 sqrt (3i).

find a linear combination for gcd(259,313). use extended euclidean algorithm.

what is inverse of 259 in z subscript 313?

what is inverse of 313 in z subscript 259?