Grant: Discovery Grants Program - Individual

In my project I intend to work towards making contributions to Number Theory, more specifically to the area of Cyclotomic Fields. 1. Let p be an odd prime number, n>0, z a primitive p^n-th root of 1 and b=p^(1/p^n). Let Qp be the field of p-adic numbers, K=Qp(z,b) and P the prime ideal of its ring of integers. The field extension K/Qp is totally ramified. After a question posed by Adrian Iovita, I intend to search for a uniformizing parameter for K/Qp; that is, an element t in K such that P=(t). Later on, I intend to work in the more general problem, stated by Filippo Viviani, of finding a uniformizing parameter for L/Qp, where L=Qp(z,a^(1/p^n)), and a is an integer such that either p does not divide a or p divides exactly a. 2. We are interested in finding families of cyclic monic polynomials of degree m>2 with integer coefficients and constant terms 1 or -1. Those families generalize a quintic family of cyclic polynomials found by Emma Lehmer and used by Schoof an Washington to construct a real p-cyclotomic field with class number divisible by a large prime. We found some ways to construct such kind of families of polynomials and calculated some interesting examples. We believe that with a more efficient implementation of our algorithms, and the use of faster computers, we can find many new examples. 3. Let p be an odd prime, z a primitive p-th root of 1, A the Sylow p-subgroup of the ideal class group of Q(z), G the Galois group of Q(z)/Q and w the Teichmuller character of G with values in Zp. We found some criteria to know when, for n and t odd, with e=(p-1)/n small, the w^(p-nt) component of A is trivial. This is related to Vandiver's conjecture. Recently Robert Osburn used our results to give an alternative proof of the fact that, when p=3 mod 4, the w^((p+1)/2) component of A is trivial. There e=2. We will work in extending that result to other values of e.