The Hasse norm theorem and a localglobal principle for multinorms Open Access
Alamoudi, Yazan (Spring 2021)
Published
Abstract
While a localglobal principle for norms from cyclic extensions of number fields is the classical Hasse Norm Theorem, such a localglobal principle fails for noncyclic extensions in general. There has been a host of results in the direction of a localglobal principle for multinorms, namely norms from a finite product of finite separable field extensions, the socalled étale algebras. In this thesis, we prove the following.
Theorem: Let Ek be a dihedral extension of degree 2n. Let E_i, 1 ≤ i ≤ n be the n distinct subextensions of E of degree n which are fixed fields under the reflections. Let L = Π_{1≤i≤n} E_i and N_{L/k} the norm from the étale algebra L to k. Then an element c ∈ k is a value of N_{L/k} if it is locally a norm at all places of k.
The proof is via the use of the Hasse norm theorem. We give an exposition of some of the relevant class field theory results leading to the Hasse Norm Theorem in the thesis. In addition, we also prove a weak approximation result as a consequence of the above theorem.
Table of Contents
1 Introduction 1
1.1 Setting and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Some results on the cohomology of groups 4
2.1 Homology, cohomology and Tate groups . . . . . . . . . . . . . . . . 4
2.2 Tate’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Further preliminaries 16
3.1 Some results from local class field theory . . . . . . . . . . . . . . . . 16
3.2 The idèles and some facts about local fields . . . . . . . . . . . . . . . 21
4 The first inequality 26
4.1 The statement and the plan . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 A relevant result on certain lattices . . . . . . . . . . . . . . . . . . . 26
4.3 The computation of the Herbrand quotients . . . . . . . . . . . . . . 28
4.4 Some implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 The second inequality for cyclic extensions and the Hasse norm theorem 33
5.1 The statement and the plan . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 the reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 The proof of the reduced case . . . . . . . . . . . . . . . . . . . . . . 35
5.4 An implication and the Hasse Norm . . . . . . . . . . . . . . . . . . . 39
6 The multinorm application 41
6.1 The multinorm result . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Weak approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Appendix A Kummer theory 47
Bibliography 49
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