Hi can you help me with these questions. I'm not sure if I done them correctly. The book that we're using is Called History of Mathematics: An Introduction. 3rd edition by Victor J Katz. At the bottom is online link to the book with notes/short examples and questions. When you copy and paste the link in another browser it will take you straight to the online book.

https://edisciplinas.usp.br/pluginfile.php/6075667/mod_resource/content/1/Victor%20J.%20Katz%20-%20A%20History%20of%20Mathematics-Pearson%20%282008%29.pdf

To help answer question five you can turn to page 668 - in chapter 19

To help answer question six you can turn to page 675-677 in chapter 19

To help answer question seven you can turn to page 677-682 in chapter 19

To help answer question eight you can turn to page 689- 694 in chapter 20

6) Chapter 19, Question 19: Determine the three roots x;, Xa, X3 of x 6x 9 = (. Use Lagrange's procedure to find the sixth-degree equation satisfied by y, where x =y + 2/y. Determine all six solutions 1 of this equation and express each explicitly as 3(x" + wx" + w2x'"), where (, x", x') is a permutation of (x1, X2, x5) and w is a complex root of x* 1 = 0. (4 points) 2 g a) Substitute for x using x = y + ;and multiply both sides of the equation by y* to create a 6!" degree equation: b) Substitute r = y? and solve the resulting quadratic equation in r. c) Given the 2 solutions, ry and r, of the quadratic equation, the 6 solutions of the 6 degree equation 1+v=3 are \ y, wiry, w3y, Vra, wyrs, @i, where w = . Find the 3 solutions to the original equation (in terms of w) using x1 = \\1] + {72, x2 = wir] + @273 and x3 = w1 +w\\Tr 7) Chapter 19, Question 25 (modified). (5 points) a)Find the distinct residues 1, a, B, . .. of 1, 5,54, 54, ... modulo 13. b) You should find that 2 is a nonresidue of the sequence of powers of 5. Determine the coset 2, 2a, 2, . and show that every member of the coset is also a nonresidue of the sequence. (A member of the coset that exceeds 13, say 13 + k, will be a nonresidue if and only if k is a nonresidue. For example, if there is no power of 5 such that 5* = 7 (mod 13), then 7 is a nonresidue and 20, 33, ... would also be nonresidues since 20 = 7 (mod 13). c) Specify p(13) and n = the number of residues of the sequence. Show that (13) = mn and give the value of m. 8) Chapter 20: Question 2: Given the hypothesis of the acute angle, both Saccheri and Lambert showed that the sum of the angles of any triangle is less than two right angles. Let the difference between 180- and the angle sum of a triangle be the defect of the triangle. Suppose triangle ABC is split into two triangles by line BD (Fig. 20.16). Show that the defect of triangle ABC is equal to the sum of the defects of triangles ABD and BDC. 180 sum of the angles of AABC = 180" sum of the angles of AABD + 180 sum of the angles of ABDC = 360 sum of the angles of AABD and ABDC. (4 points) 5) Chapter 19, Question 16: Consider the following system of linear equations given by Euler: x + 13y - 14z + 15v + 16 = 0 5x + 7y - 4z + 3v - 24 = 0 2x - 3y + 5z - 6v - 20 = 0 3x + 10y - 9z + 9v - 4 = 0. Show that these four equations are "worth only two," so that they do not determine a unique 4-tuple as a solution. There are many ways to attack this, but using the first equation to eliminate x in the 3 remaining equations works out very nicely. (5 points)