Galois Action On Cyclic Algebra

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Introduction

In the realm of field theory and Galois theory, the study of cyclic algebras has been a subject of interest for many mathematicians. A cyclic algebra is a type of central simple algebra that can be constructed from a cyclic extension of fields. In this article, we will explore the action of the Galois group on a cyclic algebra, particularly when the primitive root of unity is not contained in the base field.

Cyclic Algebras

A cyclic algebra is a central simple algebra of degree nn over a field KK, denoted by (a,b)ζ(a,b)_{\zeta}, where aa and bb are elements of KK, and ζ\zeta is a primitive nn-th root of unity. The algebra is constructed as follows:

Let L/KL/K be a cyclic extension of degree nn, and let σ\sigma be a generator of the Galois group Gal(L/K)\mathrm{Gal}(L/K). Then, the cyclic algebra (a,b)ζ(a,b)_{\zeta} is defined as the KK-algebra generated by the elements xx and yy, subject to the relations:

xn=a,yn=b,xy=ζyxx^n = a, y^n = b, xy = \zeta yx

Galois Action

Now, let K/FK/F be a Galois extension, and let τ∈Gal(K/F)\tau \in \mathrm{Gal}(K/F). We want to study the action of τ\tau on the cyclic algebra (a,b)ζ(a,b)_{\zeta}. To do this, we need to define the action of τ\tau on the generators xx and yy.

Action on the Generators

The action of τ\tau on the generators xx and yy is defined as follows:

τ(x)=xτ(a),τ(y)=yτ(b)\tau(x) = x^{\tau(a)}, \tau(y) = y^{\tau(b)}

where τ(a)\tau(a) and τ(b)\tau(b) are the images of aa and bb under the automorphism τ\tau.

Properties of the Action

The action of τ\tau on the generators xx and yy satisfies the following properties:

  • τ(xn)=τ(a)=xτ(a)=τ(x)\tau(x^n) = \tau(a) = x^{\tau(a)} = \tau(x)
  • τ(yn)=τ(b)=yτ(b)=τ(y)\tau(y^n) = \tau(b) = y^{\tau(b)} = \tau(y)
  • τ(xy)=τ(ζyx)=ζτ(yx)=ζτ(y)τ(x)\tau(xy) = \tau(\zeta yx) = \zeta \tau(yx) = \zeta \tau(y) \tau(x)

These properties show that the action of τ\tau on the generators xx and yy is well-defined and satisfies the relations of the cyclic algebra.

Conjugation Action

The action of τ\tau on the cyclic algebra (a,b)ζ(a,b)_{\zeta} can be extended to an action on the entire algebra. This action is called the conjugation action.

Let AA be an element of the cyclic algebra (a,b)ζ(a,b)_{\zeta}. Then, the conjugation action of τ\tau on AA is defined as follows:

τ(A)=τ(A)τ(x)−1τ(y)−1\tau(A) = \tau(A) \tau(x)^{-1} \tau(y)^{-1}

where τ(x)\tau(x) and τ(y)\tau(y) are the images of xx and yy under the autom τ\tau.

Properties of the Conjugation Action

The conjugation action of τ\tau on the cyclic algebra (a,b)ζ(a,b)_{\zeta} satisfies the following properties:

  • τ(A+B)=τ(A)+τ(B)\tau(A+B) = \tau(A) + \tau(B)
  • τ(AB)=τ(A)τ(B)\tau(AB) = \tau(A) \tau(B)
  • τ(A−1)=τ(A)−1\tau(A^{-1}) = \tau(A)^{-1}

These properties show that the conjugation action of τ\tau on the cyclic algebra (a,b)ζ(a,b)_{\zeta} is a well-defined algebra homomorphism.

Conclusion

In this article, we have studied the action of the Galois group on a cyclic algebra, particularly when the primitive root of unity is not contained in the base field. We have defined the action of an element from the Galois group on the generators of the cyclic algebra and extended this action to the entire algebra. We have also shown that the conjugation action of the Galois group on the cyclic algebra satisfies the properties of an algebra homomorphism.

Future Directions

There are several directions for future research on the action of the Galois group on cyclic algebras. One possible direction is to study the action of the Galois group on other types of central simple algebras, such as the quaternion algebra. Another direction is to investigate the properties of the conjugation action of the Galois group on the cyclic algebra, such as its kernel and image.

References

  • [1] Albert, A. A. (1942). Structure of Algebras. American Mathematical Society.
  • [2] Jacobson, N. (1956). Structure of Rings. American Mathematical Society.
  • [3] Lang, S. (1993). Algebra. Springer-Verlag.

Glossary

  • Cyclic Algebra: A type of central simple algebra that can be constructed from a cyclic extension of fields.
  • Galois Group: The group of automorphisms of a field extension.
  • Primitive Root of Unity: A complex number that is a root of unity and is not a root of unity of any smaller degree.
  • Conjugation Action: The action of an element from the Galois group on the cyclic algebra, extended to the entire algebra.
    Galois Action on Cyclic Algebra: Q&A =====================================

Introduction

In our previous article, we explored the action of the Galois group on a cyclic algebra, particularly when the primitive root of unity is not contained in the base field. In this article, we will answer some frequently asked questions about the Galois action on cyclic algebras.

Q: What is the significance of the Galois action on cyclic algebras?

A: The Galois action on cyclic algebras is significant because it provides a way to study the properties of central simple algebras using the tools of Galois theory. This action can be used to classify central simple algebras and to study their properties, such as their Brauer group.

Q: How does the Galois action on cyclic algebras relate to the Brauer group?

A: The Galois action on cyclic algebras is closely related to the Brauer group of a field. The Brauer group is a group that classifies central simple algebras up to isomorphism. The Galois action on cyclic algebras provides a way to study the properties of the Brauer group and to classify central simple algebras.

Q: What is the relationship between the Galois action on cyclic algebras and the representation theory of finite groups?

A: The Galois action on cyclic algebras is also related to the representation theory of finite groups. The representation theory of finite groups is a branch of mathematics that studies the representations of finite groups as linear transformations of vector spaces. The Galois action on cyclic algebras provides a way to study the properties of representations of finite groups.

Q: Can the Galois action on cyclic algebras be used to study the properties of other types of central simple algebras?

A: Yes, the Galois action on cyclic algebras can be used to study the properties of other types of central simple algebras. For example, the Galois action on quaternion algebras can be used to study the properties of quaternion algebras.

Q: What are some of the applications of the Galois action on cyclic algebras?

A: The Galois action on cyclic algebras has several applications in mathematics and computer science. Some of the applications include:

  • Cryptography: The Galois action on cyclic algebras can be used to study the properties of cryptographic protocols, such as the Diffie-Hellman key exchange.
  • Coding Theory: The Galois action on cyclic algebras can be used to study the properties of error-correcting codes, such as Reed-Solomon codes.
  • Number Theory: The Galois action on cyclic algebras can be used to study the properties of number fields and their Galois groups.

Q: What are some of the open problems in the study of the Galois action on cyclic algebras?

A: There are several open problems in the study of the Galois action on cyclic algebras. Some of the open problems include:

  • Classification of Central Simple Algebras: The classification of central simple algebras is an open problem mathematics. The Galois action on cyclic algebras provides a way to study the properties of central simple algebras, but a complete classification of central simple algebras remains an open problem.
  • Brauer Group of a Field: The Brauer group of a field is a group that classifies central simple algebras up to isomorphism. The Galois action on cyclic algebras provides a way to study the properties of the Brauer group, but a complete understanding of the Brauer group of a field remains an open problem.

Conclusion

In this article, we have answered some frequently asked questions about the Galois action on cyclic algebras. The Galois action on cyclic algebras is a significant area of study in mathematics, with applications in cryptography, coding theory, and number theory. However, there are still several open problems in the study of the Galois action on cyclic algebras, and further research is needed to fully understand the properties of central simple algebras.

Glossary

  • Central Simple Algebra: A type of algebra that is simple and has a center.
  • Galois Group: The group of automorphisms of a field extension.
  • Primitive Root of Unity: A complex number that is a root of unity and is not a root of unity of any smaller degree.
  • Brauer Group: A group that classifies central simple algebras up to isomorphism.