How to solve a cyclic quintic in radicals?
Galois theory tells us that
can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be , following Gauss we can split the problem into solving quintics and quadratics by looking at subgroups of the roots. Since 2 is a generator of the group [2,4,8,5,10,9,7,3,6,1] we can partition into the five subgroups of conjugate pairs [2,9],[4,7],[8,3],[5,6],[10,1].
Once one has one easily gets . It's easy to find . The point is that takes to and so takes to . Thus can be written down in terms of rationals (if that's your starting field) and powers of . Alas, here is where the algebra becomes difficult. The coefficients of powers of in are complicated. They can be expressed in terms of a root of a "resolvent polynomial" which will have a rational root as the equation is cyclic. Once one has done this, you have as a fifth root of a certain explicit complex number. Then one can express the other in terms of . The details are not very pleasant, but Dummit skilfully navigates through the complexities, and produces formulas which are not as complicated as they might be. Alas, I don't have the time nor the energy to provide more details.
Rotation matrix to construct canonical form of a conic
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial
The eigenvalue are , with associated eigenvectors . Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that ):
Then , and after some computations I find the canonical form
Random-digit-dialing telephone surveys used to exclude cell phone numbers. If the opinions of people who have only cell phones differed from those of people who have landline service, the poll results may not represent the entire adult population. The Pew Research Center interviewed separate random samples of cell-only and landline telephone users who were less than 30 years old and asked them to describe their political party affiliation.
or goodness of fit, homogeneity or independence?
A penny of mass 3.1 g rests on a small 29.1 g block supported by a spinning disk of radius 8.3 cm. The coefficients of friction between block and disk are 0.742 (static) and 0.64 (kinetic) while those for the penny and block are 0.617 (static) and 0.45 (kinetic).
Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is 2.2. What is the probability that there will be) at most 2 such accidents in the next 2 months?
Determine whether the given set S is a subspace of the Vector space V.
(there are multiple)
A wallet containing five 100-peso bills, four 200-peso bills, one 500-peso bill, and seven 1,000-peso bills. Which of the following probability distribution table is the appropriate to describe the given situation? *