标题： 关于分裂域和簇塔的极小生成集 显示英文标题

标题： On Minimal Generating Sets of Splitting Field and Cluster Towers

摘要： 簇塔的概念由第二作者和克里蒂卡在[4]中引入，并提出了一个问题，该问题由第一作者和巴格瓦特在[1]中回答。在本文中，我们介绍了分裂域的极小生成集的概念，并将其与簇塔的概念联系起来。我们证明了存在无穷多个有理数上的不可约多项式，对于这些多项式，它们的分裂域具有两个极端的极小生成集（一个具有给定的基数，另一个具有最小基数），并且我们有两个极端的簇塔（一个具有给定的长度，另一个具有最短长度）。我们证明了与我们在定理证明中构造的极小生成集相关的簇塔的有趣性质，从而得出结论：即使在使用极小生成集的情况下，度序列也依赖于排序。我们还建立了有理数上某一族多项式的集合成为最小极小生成集的等价条件，并计算了最小极小生成集的总数。 显示英文摘要

摘要： The concept of cluster towers was introduced by the second author and Krithika in [4] along with a question which was answered by the first author and Bhagwat in [1]. In this article we introduce the concept of minimal generating sets of splitting field and connect it to the concept of cluster towers. We establish that there exist infinitely many irreducible polynomials over rationals for which the splitting field has two extreme minimal generating sets (one of given cardinality and other of minimum cardinality) and for which we have two extreme cluster towers (one of given length and other of minimum length). We prove interesting properties of cluster tower associated with minimal generating set that we constructed in proof of the theorem and as a consequence get that degree sequence depends on the ordering even when we work with minimal generating set. We also establish an equivalent condition for a set to be minimum minimal generating set for a certain family of polynomials over rationals and count the total number of minimum minimal generating sets.