Representations Of Z N \mathbb{Z}_n Z N ​ Over Q \mathbb{Q} Q

Introduction

In the realm of abstract algebra, representation theory plays a vital role in understanding the structure of groups and their properties. The study of representations of a group over a field involves finding homomorphisms from the group to the general linear group of the field. In this article, we will delve into the representations of $\mathbb{Z}_n$, the cyclic group of order $n$, over the field of rational numbers $\mathbb{Q}$.

Background

Before we dive into the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$, let's briefly review the known results for $\mathbb{Z}_n$ over $\mathbb{C}$. It is well-known that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{C}$ sending $1$ to the roots of unity. These representations are one-dimensional and are given by the map $\sigma_k: \mathbb{Z}_n \to \mathbb{C}$, where $\sigma_k(i) = \zeta_k^i$, and $\zeta_k$ is a primitive $k$-th root of unity.

Representations of $\mathbb{Z}_n$ over $\mathbb{Q}$

Now, let's consider the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$. Since $\mathbb{Q}$ is a subfield of $\mathbb{C}$, we can use the known results for $\mathbb{Z}_n$ over $\mathbb{C}$ as a starting point. However, we need to be careful when extending the scalars from $\mathbb{C}$ to $\mathbb{Q}$, as this can lead to the loss of irreducibility.

Theorem 1

Let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $\mathbb{Q}$ be the field of rational numbers. Then, the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}$ sending $1$ to the roots of unity.

Proof

To prove this theorem, we need to show that the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the same as the representations of $\mathbb{Z}_n$ over $\mathbb{C}$. Let $\rho: \mathbb{Z}_n \to \mathbb{Q}$ be an irreducible representation. Then, we can extend the scalars to $\mathbb{C}$ by composing $\rho$ with the inclusion map $\mathbb{Q} \hookrightarrow \mathbb{C}$. This gives us a representation $\rho': \mathbb{Z}_n \to \mathbb{C}$, which is also irreducible.

Since the representations of $\mathbb{Z}_n$ over $\mathbb{C}$ are well-known, we can use the fact that $\rho'$ is irreducible to conclude that it is one of the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{C}$ sending $1$ to the roots of unity. Therefore, $\rho$ is also one of the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}$ sending $1$ to the roots of unity.

Corollary 1

Let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $\mathbb{Q}$ be the field of rational numbers. Then, the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ is equal to the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$, which is $n$.

Proof

This corollary follows directly from Theorem 1, which states that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the same as the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$. Therefore, the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ is equal to the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$, which is $n$.

Conclusion

In this article, we have shown that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the same as the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$. This result has important implications for the study of representation theory, as it allows us to use the known results for $\mathbb{Z}_n$ over $\mathbb{C}$ to study the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$. We hope that this article will be a useful resource for researchers in the field of abstract algebra and representation theory.

References

• [1] Serre, J.-P. (1977). Linear representations of finite groups. Springer-Verlag.
• [2] Fulton, W., & Harris, J. (1991). Representation theory: A first course. Springer-Verlag.
• [3] Artin, E. (1947). Galois theory. Notre Dame Mathematical Lectures, No. 2.

Future Work

There are several directions for future research in this area. One possible direction is to study the representations of $\mathbb{Z}_n$ over other fields, such as the field of real numbers or the field of $p$-adic numbers. Another possible direction is to study the representations of other groups, such as the symmetric group or the general linear group.

Acknowledgments

Q: What is the significance of the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$?

A: The representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are significant because they provide a way to study the structure of the cyclic group $\mathbb{Z}_n$ over the field of rational numbers $\mathbb{Q}$. This is important because it allows us to understand the properties of $\mathbb{Z}_n$ in a more general setting than just over the complex numbers.

Q: How do the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ relate to the representations of $\mathbb{Z}_n$ over $\mathbb{C}$?

A: The representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the same as the representations of $\mathbb{Z}_n$ over $\mathbb{C}$. This means that the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ is equal to the number of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$, which is $n$.

Q: What are the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$?

A: The irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}$ sending $1$ to the roots of unity.

Q: How do you extend the scalars from $\mathbb{C}$ to $\mathbb{Q}$?

A: To extend the scalars from $\mathbb{C}$ to $\mathbb{Q}$, you can compose the representation $\rho: \mathbb{Z}_n \to \mathbb{C}$ with the inclusion map $\mathbb{Q} \hookrightarrow \mathbb{C}$. This gives you a representation $\rho': \mathbb{Z}_n \to \mathbb{Q}$.

Q: What are the implications of this result for the study of representation theory?

A: This result has important implications for the study of representation theory. It allows us to use the known results for $\mathbb{Z}_n$ over $\mathbb{C}$ to study the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$. This is useful because it provides a way to study the properties of $\mathbb{Z}_n$ in a more general setting than just over the complex numbers.

Q: What are some possible directions for future research in this area?

A: Some possible directions for future research in this area include studying the representations of $\mathbb{Z}_n$ over other fields, such as the field of real numbers or the field of $p$-adic numbers. Another possible direction is to study the representations of other groups, such as the symmetric group or the general linear group.

Q: What are some of the key concepts and techniques used in this area of research?

A: Some of the key concepts and techniques used in this area of research include group theory, representation theory, and Galois theory. These concepts and techniques are used to study the properties of the cyclic group $\mathbb{Z}_n$ and its representations over different fields.

Q: What are some of the applications of this research?

A: Some of the applications of this research include understanding the properties of the cyclic group $\mathbb{Z}_n$ and its representations over different fields. This is useful in a variety of areas, including number theory, algebraic geometry, and cryptography.

Q: What are some of the challenges and open problems in this area of research?

A: Some of the challenges and open problems in this area of research include studying the representations of $\mathbb{Z}_n$ over other fields, such as the field of real numbers or the field of $p$-adic numbers. Another challenge is to study the representations of other groups, such as the symmetric group or the general linear group.

Q: What are some of the resources and references that are useful for learning more about this area of research?

A: Some of the resources and references that are useful for learning more about this area of research include the books "Linear Representations of Finite Groups" by Jean-Pierre Serre and "Representation Theory: A First Course" by William Fulton and James Harris. These books provide a comprehensive introduction to the subject and are a good starting point for further study.