Cyclotomic fields

在我的项目中，我打算致力于对数论做出贡献，更具体地说，是在割圆领域1。设p是奇素数，n&gt；0，z是1的本原p^n次根，b=p^(1/p^n)。设Qp是p-进数域，K=Qp(z，b)，P是它的整数环的素理想。场扩张K/Qp是完全分支的。在禤浩焯提出的一个问题之后，我打算寻找K/Qp的一个一致参数，即K中使P=(T)的元素t。稍后，我打算解决菲利波·维维亚尼所说的更一般的问题，找到L/qp的统一参数，其中L=qp(z，a^(1/p^n))，a是一个整数，使得p不能除以a，或者p恰好除以a.2。我们感兴趣的是寻找具有整数系数和常数项1或-1的m&gt；2次循环一次多项式族。这些族推广了Emma Lehmer发现的五次循环多项式族，并被华盛顿的Schoof用来构造一个类数可被大素数整除的实p-分圆域。我们找到了构造这类多项式族的一些方法，并计算了一些有趣的例子。我们相信，随着我们算法的更有效实施，以及速度更快的计算机的使用，我们可以找到许多新的例子。设p是奇素数，z是1的本原p次根，A是q(Z)的理想类群的Sylow p-子群，G是q(Z)/q的Galois群，w是G的取值于Zp的TeichMuller特征标.我们发现了一些判定准则，对于n和t奇数，当e=(p-1)/n小时，A的w^(p-Nt)分量是平凡的。这与Vandiver的猜想有关。最近，Robert Osburn用我们的结果给出了另一种证明，证明了当p=3mod 4时，A的w^((p+1)/2)分量是平凡的。这里e=2。我们将把这个结果推广到e的其他值。

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.