This report concentrates on two categories of quintics that create diverse problems for dealing with them. The 1st family is a popular group of quintics which are known as Emma Lehmer’s Quintics. These quintics are recognized to contain the cyclic group of order 5 as the Galois group and one may expect that expressing the roots in terms of radicals will give easy expressions from which Emma Lehmer’s polynomials may be retrieved. However, we reveal that the expresions of the roots in terms of radicals is a lot more complex than anticipated. We also look at the simpple equation f(x)=x^5+ax+p and show that, for a fixed nonzero integer p, the polynomial f is solvable by radicals only for finitely many integers a. David Dummit in Solving Solvable Quintics gives a powerful method that enables us to identify when a quintic is solvable and also to solve for its roots. We’ll use Dummit’s approach to examine the two families of quintics.
1 Summary of Dummit’s method
2 Application of Dummit’s Method to Emma Lehmer’s Quintics
2.1 Computation of the li with Mathematica
2.2 Proof the li are the right ones
2.3 Computation of θ in terms of n 3 f(x) = x5 + ax + p 15
3.1 When is f irreducible?
3.2 Is f solvable? Dummit’s method
3.3 Method of looking at the number of complex roots of f to determine whether f is solvable…..
Source: University of Maryland