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*.

*Contents*

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