Rank of P\'olya Groups in Lecacheux Parametric Family of Quintic Fields

In this article, we study the Pólya group of a new family of quintic fields, namely Lecacheux quintic fields.

We show that the associated Pólya groups can be arbitrarily large elementary abelian \(5\)-groups.

Using density arguments, we prove that for every positive integer $k$, the set of odd integers $s$ such that the $5-$rank of the Pólya group of the corresponding Lecacheux quintic field is at least $k$ has a positive density.

Combining this with a result of Golod and Shafarevich, we see that for a positive proportion of $s$, the corresponding Lecacheux quintic fields admit an infinte $5-$class field tower.

We also establish an upper bound for the Pólya numbers of these fields in terms of the orders of their corresponding Pólya groups.

In addition, we prove that several fields in this family are non-monogenic despite having index one.