A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem.

# A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem.

Pavel Solomatin
p.solomatin@math.leidenuniv.nl
Leiden University, Mathematical Department,
Niels Bohrweg 1, 2333 CA Leiden
Den Haag, 2019
###### Abstract

Given a number field one associates to it the set of Dedekind zeta-functions of finite abelian extensions of . In this short note we present a proof of the following Theorem: for any number field the set determines the isomorphism class of . This means that if for any number field the two sets and coincide, then . As a consequence of this fact we deduce an alternative approach towards the proof of Neukirch-Uchida Theorem for the case of non-normal extensions of number fields.

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Acknowledgements: I would like to express my gratitude to professor Bart de Smit for all of his efforts and exciting discussions during the project.

## 1 Introduction

### 1.1 Motivation

Let be a number field, i.e. a finite extension of the field of rational numbers and let be the absolute Galois group of , that is for some fixed algebraic closure of . The isomorphism class of captures a lot of important information about arithmetical properties of . The famous Neukirch-Uchida Theorem states that for given number fields the existence of a topological isomorphism of pro-finite groups implies the existence of an isomorphism of fields themselves. In 1969 Neukirch  gave a proof for the case of normal extensions of . He proved this by recovering Dedekind zeta-function of from in group-theoretical terms and then applying the famous Chebotarev density argument to show that in this case determines the isomorphism class of . A few years later in 1976 Uchida  extended Neukirch’s results to the case of arbitrary number fields. Uchida’s approach was then also used by himself and others to generalise the Theorem to the case of all global fields. For a modern exposition see Chapter XII in . Without any doubt Uchida’s proof is beautiful and important, but it contains some difficult technical details which make this proof a bit less clear especially for those who are relatively new to the topic. The goal of the present note is to provide an alternative, in some sense more elementary approach to the proof of Uchida’s part. The new proof also has another advantage, since it stays closer to Neukirch’s original idea. This new approach is based on the following idea.

Given a number field we associate to it a set of Dedekind zeta-functions of finite abelian extensions of :

 ΛK={ζL(s) | L is a finite abelian extension of K}.

Our main goal is to prove the following Theorem:

###### Theorem 1.

For any number field the set determines the isomorphism class of . This means that if for any other number field the two sets and coincide, then .

The following observation shows that Theorem 1 allows us to achieve our goal and produce an alternative way to Uchida’s part:

###### Corollary 1.

In the above settings suppose that . Then and therefore .

###### Proof.

Indeed, given an isomorphism class of we consider all closed subgroups of finite index such that the quotient is a finite abelian group. By pro-finite Galois theory we have one-to-one correspondence between such and finite abelian extensions of within fixed algebraic closure given by . Now by using Neukirch’s Theorem, see chapter 4 in  we reconstruct in a group theoretical manner from and therefore reconstruct from . ∎

From now on we concentrate our attention towards the proof of 1.

### 1.2 On the Proof of Theorem 1

The inspiration behind Theorem 1 is the following result due professor Bart de Smit:

###### Theorem 2.

For each number field there exists an abelian extension of degree three and a character of such that occurs only for the isomorphism class of the field , i.e. if for any other number field and any abelian extension of there exists a character of such that then .

###### Proof.

See Theorem 10.1 from . ∎

To deduce Theorem 1 we extend Theorem 2 by replacing the L-function of the abelian character by the Dedekind -function of the abelian extension of :

###### Theorem 3.

For each number field there exists an abelian extension of degree three such that pair , occurs only for the isomorphism class of the field , i.e. if for any other number field and any abelian extension of we have and  =  then .

###### Remark 1.

Note that the degree of a number field is determined by . Therefore, can be recovered from as unique element whose corresponding field has minimal degree.

The above remark shows that Theorem 1 and Theorem 3 are equivalent and we can focus on proving the last statement.

The article has the following structure. In the next section we recall some basic group theoretical settings of Theorem 2 and then provide a proof for a generalisation of these settings, see Theorem 5. After that we will deduce Theorem 3 from Theorem 5.

## 2 The proof

### 2.1 Group Theoretical Preliminaries

Let be a finite group, a subgroup of index , and be a cyclic group of order , where is an odd prime. Consider the -set of left cosets. We fix some representatives of such that is a coset corresponding to the group . Let us regard semi-direct products and , where acts on the components of by permuting them as the cosets . Let be the homomorphism from to the group defined on the element as i.e. is the projection on the first coordinate. This is indeed a homomorphism because every fixes the first coset of . In this setting the following holds:

###### Theorem 4 (Bart de Smit).

For any subgroup and any character : if then and are conjugate in .

###### Proof.

See Theorem 7 from . ∎

To apply this result we fix and first prove the following auxiliary statement:

###### Lemma 1.

The induced representation is an irreducible representation of .

###### Proof.

In order to verify irreducibility of we regard the standard scalar product and show that . Applying Frobenius reciprocity:

 (Ind~G~H(χ),Ind~G~H(χ))~G=(χ,Ind~G~H(χ)|~H)~H=1|~H|∑~h∈~H¯χ(~h)⋅Tr(Ind~G~H(χ)|~H(~h)). (1)

Let then by definition of we have . Now consider the matrix . We fix the following representatives for cosets of as , where are representative of cosets of we picked before. By definition of the induced representation and because fixes first conjugacy class of we have that in the top left corner of that matrix is located. Now we fix an integer and consider diagonal element on -th position. Regard the permutation of cosets by and denote by an index such that . If then and therefore such adds no contribution to the expression 1. Otherwise, by definition of the induced representation we have for some index . In other words, is an index such that . For fixed and there are elements , such that pairwise coincide in all coordinates except the -th one. Because we have that sum of for those , , is zero and because they coincide on first coordinate we have for in . Therefore for fixed we have:

 ∑~h∈~H¯χ(~h)ai(~h)=0.

Now we regard expression 1:

 1|~H|∑~h∈~H¯χ(~h)⋅Tr(Ind~G~H(χ)|~H(~h))=1|~H|∑~h∈~H¯χ(~h)⋅(χ(~h)+i≤n∑i>1ai(~h))= =1|~H|∑~h∈~H¯χ(~h)χ(~h)+1|~H|∑~h∈~H(¯χ(~h)⋅(i≤n∑i>1ai(~h)))= =1|~H|∑~h∈~H1+1|~H|∑i>1(∑~h∈~H¯χ(~h)ai(~h))=1+0.

By using this lemma we can prove the main group theoretical result of this note:

###### Theorem 5.

In the above settings let and let . Suppose that and . Then either or .

###### Proof.

Since we have that has only three characters and therefore:

 Ind~GU~H,χ(1)≃Ind~G~H(χ)⊕Ind~G~H(¯χ)⊕Ind~G~H(1).

Hence, from the assumption of the Theorem it follows that:

 Ind~G~H(χ)⊕Ind~G~H(¯χ)≃Ind~G~H′(χ′)⊕Ind~G~H′(¯χ′).

In lemma 1 we showed that , are irreducible representation of . But if a direct sum of two irreducible representations of a finite group is isomorphic to a direct sum of two other non-zero representations then those representations are pairwise isomorphic up to swap. It follows that either or . ∎

## 3 Recovering Number Fields from ζ-functions

###### Proof of Theorem 3.

Suppose is a number field such that does not determine i.e. there exists a number field such that , but . Then according to the well-known Theorem of Perlis  this means that the normal closure of contains and there exists a non-trivial Gassmann triple with , , .

In this settings we construct a Galois extension of containing and such that the Galois group is and , for , , as in Theorem 4. This is possible due to Proposition 9.1 from . See the diagram below:

{tikzcd}

& M \arrow[rdddd, dash,bend left = 10, ”~H’”] \arrow[ldddd, dash,bend right = 70, ”~H=C3n ⋊H”’] \arrow[ddddddd, dash,bend left = 100, ”~G = C3n ⋊G”] \arrow[d, dash]

& LN \arrow[d, dash] \arrow[ldd, dash ]

& N \arrow[ldd, dash, ”H”’ ] \arrow[rdd, dash, ”H’”] \arrow[ddddd, dash,bend right = 20, ”G”]

L\arrow[d, dash, ”C3”’ ] &

K \arrow[rddd,dash] & &K’ \arrow[lddd,dash]

& Q

Our goal is to show that occurs only for fields isomorphic to . Let be any abelian extension of such that . Then and share the same normal closure over and therefore is a subfield of . According to remark 1 we also have that degree of over is three. Observe that in notations of Theorem 5 from the previous section one has: for a non-trivial character of and for a non-trivial character of . By the induction property of Artin L-functions we have: .

Finally, because of Chebotarev density argument and because every place of is determined by the characteristic of its residue degree:

 LQ(Ind~GU~H,χ(1),s)=LQ(Ind~GU~H′,χ′(1),s)⇔Ind~GU~H,χ(1)≃Ind~GU~H′,χ′(1).

This means that from assumption of Theorem 3 we deduced conditions of Theorem 5. Therefore because of Theorem 4 we have that and are conjugate and hence is isomorphic to .

## References

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