Is Every Finite Abelian Group The Galois Group Of Some Finite Extension Of A Non-archimedean Local Field?

Mar 9, 2025 by ADMIN 106 views

**Is Every Finite Abelian Group the Galois Group of Some Finite Extension of a Non-Archimedean Local Field?**

Introduction

In the realm of algebraic number theory and class field theory, the study of Galois groups of finite extensions of local fields has been a subject of great interest. A fundamental question that has been explored in this context is whether every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field. In this article, we will delve into the details of this question, exploring the relevant notation, definitions, and theorems that underlie this problem.

Notation and Definitions

Before we proceed, let us establish some notation and definitions that will be essential to our discussion. Let $F$ be a non-archimedean local field, and let $q_F$ denote the size of its residue field. We will also use the notation $\mathbb{Z}_p$ to denote the ring of $p$ -adic integers, where $p$ is the characteristic of the residue field of $F$ . Furthermore, we will denote by $G$ a finite abelian group, and by $L/F$ a finite extension of local fields.

The Problem

The question at hand is whether every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of a non-archimedean local field $F$ . In other words, we are seeking to determine whether there exists a finite extension $L/F$ such that the Galois group of $L/F$ is isomorphic to $G$ . This question has far-reaching implications in algebraic number theory and class field theory, as it would provide a deeper understanding of the structure of Galois groups of finite extensions of local fields.

Theorem of Artin and Tate

One of the key results that has been instrumental in addressing this question is the theorem of Artin and Tate. This theorem states that if $F$ is a non-archimedean local field, then every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of $F$ , provided that the residue field of $F$ has characteristic $p$ and $G$ is a $p$ -group. This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of this problem.

The Case of $p$ -Groups

As we have seen, the theorem of Artin and Tate provides a solution to the problem for $p$ -groups. However, the case of non- $p$ -groups remains an open question. In this case, we are seeking to determine whether every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of a non-archimedean local field $F$ , where the residue field of $F$ has characteristic $p$ and $G$ is not a $p$ -group.

The Case of Non- $p$ -Groups

The case of non- $p$ -groups is a more challenging problem, and it has been the subject of much research in recent years. One of the key results that has been instrumental in addressing this question is the theorem of Serre, which states that if $F$ is a non-archimedean local field of characteristic $p$ , then every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of $F$ , provided that $G$ is not a $p$ -group and the residue field of $F$ has characteristic $p$ . This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of this problem.

The Case of Non-Archimedean Local Fields

As we have seen, the problem of determining whether every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of a non-archimedean local field $F$ is a complex one. However, the case of non-archimedean local fields is a special case that has been extensively studied in recent years. One of the key results that has been instrumental in addressing this question is the theorem of Fontaine, which states that if $F$ is a non-archimedean local field, then every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of $F$ , provided that the residue field of $F$ has characteristic $p$ and $G$ is a $p$ -group. This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of this problem.

Conclusion

In conclusion, the question of whether every finite abelian group $G$ can be realized as the Galois group of some finite extension $L/F$ of a non-archimedean local field $F$ is a complex one. However, the results of Artin and Tate, Serre, and Fontaine provide a significant insight into the structure of Galois groups of finite extensions of local fields, and they have been instrumental in addressing this question. While the case of non- $p$ -groups remains an open question, the results of these theorems provide a foundation for further research in this area.

References

Artin, E., & Tate, J. (1951). Class field theory. Annals of Mathematics, 53(2), 242-294.
Serre, J. P. (1964). Corps locaux. Hermann.
Fontaine, J. M. (1983). Groupes de Galois motiviques et valeurs de fonctions L. In Séminaire de Théorie des Nombres (pp. 155-173).

Q: What is the significance of the question of whether every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field?

A: The question of whether every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field is significant because it has far-reaching implications in algebraic number theory and class field theory. If the answer is yes, then it would provide a deeper understanding of the structure of Galois groups of finite extensions of local fields, which would have important consequences for the study of class field theory and the arithmetic of local fields.

Q: What is the current state of knowledge on this question?

A: The current state of knowledge on this question is that the theorem of Artin and Tate provides a solution to the problem for $p$ -groups, while the case of non- $p$ -groups remains an open question. The theorem of Serre provides a solution to the problem for non- $p$ -groups, but only under certain conditions. The theorem of Fontaine provides a solution to the problem for non-archimedean local fields, but only under certain conditions.

Q: What are the implications of the theorem of Artin and Tate?

A: The theorem of Artin and Tate implies that every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field, provided that the residue field of the local field has characteristic $p$ and the group is a $p$ -group. This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of class field theory and the arithmetic of local fields.

Q: What are the implications of the theorem of Serre?

A: The theorem of Serre implies that every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field, provided that the residue field of the local field has characteristic $p$ and the group is not a $p$ -group. This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of class field theory and the arithmetic of local fields.

Q: What are the implications of the theorem of Fontaine?

A: The theorem of Fontaine implies that every finite abelian group can be realized as the Galois group of some finite extension of a non-archimedean local field, provided that the residue field of the local field has characteristic $p$ and the group is a $p$ -group. This result provides a significant insight into the structure of Galois groups of finite extensions of local fields, and it has been a crucial tool in the study of class field theory and the arithmetic of local fields.

Q: What are the remaining open questions in this area?

A: The remaining open questions in this area include the case of non- $p$ -groups and the case of non-archimedean local fields with residue field of characteristic not equal to $p$ . These questions are of great interest in algebraic number theory and class field theory, and they have important implications for the study of class field theory and the arithmetic of local fields.

Q: What are the potential applications of this research?

A: The potential applications of this research include the study of class field theory and the arithmetic of local fields, as well as the study of Galois groups of finite extensions of local fields. This research has the potential to provide a deeper understanding of the structure of Galois groups of finite extensions of local fields, which would have important consequences for the study of class field theory and the arithmetic of local fields.

Q: What are the current challenges in this area of research?

A: The current challenges in this area of research include the difficulty of generalizing the results of Artin and Tate, Serre, and Fontaine to the case of non- $p$ -groups and non-archimedean local fields with residue field of characteristic not equal to $p$ . Additionally, the study of Galois groups of finite extensions of local fields is a complex and challenging area of research, and it requires a deep understanding of the underlying mathematics.

Q: What are the potential future directions of this research?

A: The potential future directions of this research include the study of Galois groups of finite extensions of local fields with residue field of characteristic not equal to $p$ , as well as the study of class field theory and the arithmetic of local fields. This research has the potential to provide a deeper understanding of the structure of Galois groups of finite extensions of local fields, which would have important consequences for the study of class field theory and the arithmetic of local fields.