Which gives us three linear equations with coefficients over the splitting field of x ^ 3-x-1; by inverting a matrix we can solve for p ^ n, q ^ n, r ^ n and then we can raise them to the " k " th power and compute the sum.
42.
The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial . ( In mathematical jargon, these transformations are called automorphisms . ) The Galois group of consists of two elements : The identity transformation, which sends every complex number to itself, and complex conjugation, which sends " i " to " " i ".
43.
The above construction of the Galois group for a fifth degree polynomial only applies to the " general polynomial "; specific polynomials of the fifth degree may have different Galois groups with quite different properties, e . g . has a splitting field generated by a primitive 5th root of unity, and hence its Galois group is abelian and the equation itself solvable by radicals; moreover, the argument does not provide any rational-valued quintic that has or as its Galois group.
44.
Use Hensel's lift to get the factorization over the p-adic integers, find the ring of integers A of the splitting field of the p-adic polynomial, the galois group is abelian so you can take an arbitrary factor of ( p ) as P and then look at A / P ^ r as the " extension field " of Z / p ^ rZ . Make sure that n and p are coprime ( 6 and 2 is naughty ) otherwise the factorization can change.
