On the distribution of norm groups in the intervals corresponding to odd degree extensions of algebraic number fields

Abstract

Let X be a subgroup of a group Y. The interval (X, Y) is the set of subgroups of Y that contain X including X and Y. Let K/k be a finite extension of a $\mathfrak{p}$-adic number field k. One of the fundamental theorems local class field theory establishes a correspondence between the finite number of norm groups contained in the interval $(N_{K/k}{K}^{*},k^{*})$ and finite extensions of k. In our earlier work, we proved that $N_{K/k}{K}^{*}\subseteq N_{L/k}{L}^{*}$ iff $L\subseteq K$ for any finite Galois extensions of an algebraic number field k. It is natural to determine the norm groups contained in the interval $(N_{K/k}{K}^{*},k^{*})$ for a given finite extension K/k of algebraic number fields. In our earlier work, we showed that there are extensions K/k such that the corresponding interval contains only a finite number of norm groups, and there are extensions with the corresponding interval containing infinitely many norm groups. The extensions that we considered in our earlier work were primarily of even degrees. In the present work, we investigate the distribution of norm groups in the intervals corresponding to extensions of algebraic number fields of primarily odd degrees divisible by two primes.

Keywords:

• Class field theory
• finite groups
• Galois theory
• norm groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11R32
• 11R37
• 20B05

Acknowledgements

The author is grateful to Robert Guralnick for his generous help in preparation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.