# Constructing abelian extensions with prescribed norms

@article{Frei2020ConstructingAE, title={Constructing abelian extensions with prescribed norms}, author={C. Frei and R. Richard}, journal={arXiv: Number Theory}, year={2020} }

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as norms of elements in $L$. In particular, this shows existence of such extensions for any given parameters.
Our approach relies on class field theory and a recent formulation of Tate's characterisation of the Hasse norm principle, a local-global principle for… Expand

#### References

SHOWING 1-10 OF 15 REFERENCES

Number fields with prescribed norms.

- Mathematics
- 2018

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of… Expand

The Hasse norm principle for abelian extensions

- Mathematics
- 2015

Abstract:We study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$… Expand

A survey of computational class field theory

- Mathematics
- 1999

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods… Expand

On knots in algebraic number theory.

- Mathematics
- 1979

"The automorphisms induced in the class groups by the automorphisms of the field, the properties of the norm-residues in the noncyclic cases, the passage to the limit (inductive or projective) when… Expand

An explicit bound for the least prime ideal in the Chebotarev density theorem

- Mathematics
- 2016

We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and… Expand

Computing class fields via the Artin map

- Computer Science, Mathematics
- Math. Comput.
- 2001

Based on an explicit representation of the Artin map for Kummer extensions, a method to compute arbitrary class fields in the case where the field contains sufficiently many roots of unity is presented. Expand

Beweis eines Satzes und Wiederlegung einer Vermutung über das allgemeine Normenrestsymbol

- Mathematics
- 1931

Zéro-cycles sur les espaces homogènes et problème de Galois inverse

- Mathematics
- 2018

Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th\'el\`ene, Sansuc, Kato and Saito on the image… Expand

Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental

- Principles of Mathematical Sciences],
- 1999