The $4$-rank of class groups of $K(\sqrt{n})$

Peter Koymans; Adam Morgan; Harry Smit

The $4$-rank of class groups of $K(\sqrt{n})$

Number Theory 2021-03-09 v1

Authors: Peter Koymans , Adam Morgan , Harry Smit

Abstract

Let $K/\mathbb{Q}$ be a quadratic extension. In this paper we study the $4$ -rank of the class group $\text{Cl}(K(\sqrt{n}))$ , where $n$ varies over squarefree rational integers. We show that for $100\%$ of squarefree $n$ , the $4$ -rank is given by an explicit formula involving the $2$ -rank of $\text{Cl}(K)$ and the number of prime factors of $n$ which are inert in $K/\mathbb{Q}$ .

Keywords

class group 4-rank quadratic extension squarefree integer 2-rank inert prime

Cite

@article{arxiv.2101.03407,
  title  = {The $4$-rank of class groups of $K(\sqrt{n})$},
  author = {Peter Koymans and Adam Morgan and Harry Smit},
  journal= {arXiv preprint arXiv:2101.03407},
  year   = {2021}
}

Comments

29 pages

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