Conditions Of "Norm" Isomorphism In Quaternion Algebra

Introduction

In the realm of algebra, the study of quadratic forms and their properties has been a subject of interest for mathematicians for centuries. One of the fundamental concepts in this area is the notion of a "norm" isomorphism, which plays a crucial role in understanding the structure of quaternion algebras. In this article, we will delve into the conditions under which a "norm" isomorphism exists in quaternion algebra, and explore the implications of these conditions on the properties of the algebra.

Quaternion Algebra and Quadratic Forms

Before we dive into the specifics of the "norm" isomorphism, let's briefly review the basics of quaternion algebra and quadratic forms. Quaternion algebra is a mathematical structure that extends the real numbers to include three imaginary units, denoted by $i$, $j$, and $k$. These units satisfy the following relations:

$$i^2 = j^2 = k^2 = -1$$

$$ij = k, \quad jk = i, \quad ki = j$$

$$ji = -k, \quad kj = -i, \quad ik = -j$$

A quaternion algebra over a field $F$ is a four-dimensional vector space over $F$ with a basis $\{1, i, j, k\}$, where the multiplication of two quaternions is defined as follows:

$$a_0 + a_1i + a_2j + a_3k \cdot b_0 + b_1i + b_2j + b_3k = (a_0b_0 - a_1b_1 - a_2b_2 - a_3b_3) + (a_0b_1 + a_1b_0 + a_2b_3 - a_3b_2)i + (a_0b_2 + a_2b_0 + a_3b_1 - a_1b_2)j + (a_0b_3 + a_3b_0 + a_1b_2 - a_2b_1)k$$

A quadratic form over a field $F$ is a homogeneous polynomial of degree two in a set of variables, with coefficients in $F$. In the context of quaternion algebra, a quadratic form can be represented as a quadratic polynomial in the basis elements $\{1, i, j, k\}$.

Definition of "Norm" Isomorphism

Given a quaternion algebra $A$ over a field $F$, a "norm" isomorphism is a map $\widetilde{\mathrm{N}}: A \to F$ that satisfies the following properties:

1. Homomorphism: $\widetilde{\mathrm{N}}(xy) = \widetilde{\mathrm{N}}(x)\widetilde{\mathrm{N}}(y)$ for all $x, y \in A$.
2. Non-degeneracy: $\widetilde{\mathrm{N}}(x) = 0$ if and only if $x = 0$.
3. Normalization: $\widetilde{\mathrm{N}}(1) = 1$.

The "norm" isomorphism is a crucial tool in understanding the structure of quaternion algebras, as it provides a way to associate a quadratic form with each element of the.

Conditions for the Existence of "Norm" Isomorphism

The existence of a "norm" isomorphism in a quaternion algebra depends on the properties of the algebra itself. In particular, the following conditions must be satisfied:

1. Associativity: The multiplication of quaternions must be associative, i.e., $(xy)z = x(yz)$ for all $x, y, z \in A$.
2. Distributivity: The multiplication of quaternions must be distributive over addition, i.e., $x(y + z) = xy + xz$ for all $x, y, z \in A$.
3. Commutativity: The multiplication of quaternions must be commutative, i.e., $xy = yx$ for all $x, y \in A$.
4. Non-degeneracy: The quadratic form associated with the algebra must be non-degenerate, i.e., the matrix representing the quadratic form must be invertible.

If these conditions are satisfied, then a "norm" isomorphism exists in the quaternion algebra.

Implications of the Conditions

The conditions for the existence of a "norm" isomorphism have significant implications for the properties of the quaternion algebra. In particular:

1. Isomorphism: If a "norm" isomorphism exists, then the quaternion algebra is isomorphic to the algebra of quaternions over the field $F$.
2. Quadratic Form: The quadratic form associated with the algebra is non-degenerate, and its matrix is invertible.
3. Associativity: The multiplication of quaternions is associative, and the algebra satisfies the associativity property.
4. Distributivity: The multiplication of quaternions is distributive over addition, and the algebra satisfies the distributivity property.

Conclusion

In conclusion, the conditions for the existence of a "norm" isomorphism in quaternion algebra are crucial in understanding the structure of the algebra. The existence of such an isomorphism implies that the algebra is isomorphic to the algebra of quaternions over the field $F$, and that the quadratic form associated with the algebra is non-degenerate. The conditions for the existence of a "norm" isomorphism have significant implications for the properties of the quaternion algebra, and are essential in the study of quadratic forms and their applications.

References

• Lam, T. Y. (2005). Introduction to Quadratic Forms over Fields. Springer-Verlag.
• Jacobson, N. (1980). Basic Algebra II. W.H. Freeman and Company.
• Artin, E. (1967). Galois Theory. Dover Publications.

Further Reading

For further reading on the topic of quaternion algebra and quadratic forms, we recommend the following resources:

• Lam, T. Y. (2005). Introduction to Quadratic Forms over Fields. Springer-Verlag.
• Jacobson, N. (1980). Basic Algebra II. W.H. Freeman and Company.
• Artin, E. (1967). Galois Theory. Dover Publications.

Introduction

In our previous article, we explored the conditions for the existence of a "norm" isomorphism in quaternion algebra. In this article, we will answer some of the most frequently asked questions about this topic, providing further clarification and insights into the properties of quaternion algebras.

Q: What is a "norm" isomorphism in quaternion algebra?

A: A "norm" isomorphism in quaternion algebra is a map $\widetilde{\mathrm{N}}: A \to F$ that satisfies the following properties:

1. Homomorphism: $\widetilde{\mathrm{N}}(xy) = \widetilde{\mathrm{N}}(x)\widetilde{\mathrm{N}}(y)$ for all $x, y \in A$.
2. Non-degeneracy: $\widetilde{\mathrm{N}}(x) = 0$ if and only if $x = 0$.
3. Normalization: $\widetilde{\mathrm{N}}(1) = 1$.

Q: What are the conditions for the existence of a "norm" isomorphism in quaternion algebra?

A: The conditions for the existence of a "norm" isomorphism in quaternion algebra are:

1. Associativity: The multiplication of quaternions must be associative, i.e., $(xy)z = x(yz)$ for all $x, y, z \in A$.
2. Distributivity: The multiplication of quaternions must be distributive over addition, i.e., $x(y + z) = xy + xz$ for all $x, y, z \in A$.
3. Commutativity: The multiplication of quaternions must be commutative, i.e., $xy = yx$ for all $x, y \in A$.
4. Non-degeneracy: The quadratic form associated with the algebra must be non-degenerate, i.e., the matrix representing the quadratic form must be invertible.

Q: What are the implications of the conditions for the existence of a "norm" isomorphism in quaternion algebra?

A: The conditions for the existence of a "norm" isomorphism in quaternion algebra have significant implications for the properties of the quaternion algebra. In particular:

1. Isomorphism: If a "norm" isomorphism exists, then the quaternion algebra is isomorphic to the algebra of quaternions over the field $F$.
2. Quadratic Form: The quadratic form associated with the algebra is non-degenerate, and its matrix is invertible.
3. Associativity: The multiplication of quaternions is associative, and the algebra satisfies the associativity property.
4. Distributivity: The multiplication of quaternions is distributive over addition, and the algebra satisfies the distributivity property.

Q: Can a "norm" isomorphism exist in a quaternion algebra that does not satisfy the conditions?

A: No, a "norm" isomorphism cannot exist in a quaternion algebra that does not satisfy the conditions. The conditions are necessary and sufficient for the existence of a "norm isomorphism.

Q: What are some examples of quaternion algebras that satisfy the conditions for the existence of a "norm" isomorphism?

A: Some examples of quaternion algebras that satisfy the conditions for the existence of a "norm" isomorphism include:

• The algebra of quaternions over the real numbers, $\mathbb{H}$.
• The algebra of quaternions over the complex numbers, $\mathbb{H}(\mathbb{C})$.
• The algebra of quaternions over the quaternions, $\mathbb{H}(\mathbb{H})$.

Q: What are some examples of quaternion algebras that do not satisfy the conditions for the existence of a "norm" isomorphism?

A: Some examples of quaternion algebras that do not satisfy the conditions for the existence of a "norm" isomorphism include:

• The algebra of quaternions over the rational numbers, $\mathbb{H}(\mathbb{Q})$.
• The algebra of quaternions over the p-adic numbers, $\mathbb{H}(\mathbb{Q}_p)$.
• The algebra of quaternions over the finite fields, $\mathbb{H}(\mathbb{F}_q)$.

Conclusion

In conclusion, the conditions for the existence of a "norm" isomorphism in quaternion algebra are crucial in understanding the structure of the algebra. The existence of such an isomorphism implies that the algebra is isomorphic to the algebra of quaternions over the field $F$, and that the quadratic form associated with the algebra is non-degenerate. We hope that this Q&A article has provided further clarification and insights into the properties of quaternion algebras.

References

• Lam, T. Y. (2005). Introduction to Quadratic Forms over Fields. Springer-Verlag.
• Jacobson, N. (1980). Basic Algebra II. W.H. Freeman and Company.
• Artin, E. (1967). Galois Theory. Dover Publications.

Further Reading

For further reading on the topic of quaternion algebra and quadratic forms, we recommend the following resources:

• Lam, T. Y. (2005). Introduction to Quadratic Forms over Fields. Springer-Verlag.
• Jacobson, N. (1980). Basic Algebra II. W.H. Freeman and Company.
• Artin, E. (1967). Galois Theory. Dover Publications.