Hamilton's Quaternions (W. R. HAMLLTON, 1843) We consider...

Hamilton's Quaternions (W. R. HAMLLTON, 1843) We consider the following map $$ \begin{array}{r} H: \mathbb{C} \times \mathbb{C} \longrightarrow M(2 \times 2 ; \mathbb{C}) \\ (z, w) \mapsto H(z, w):=\left(\begin{array}{cc} z & -w \\ \bar{w} & \bar{z} \end{array}\right) \end{array} $$ and denote its image by $$ \mathcal{H}:=\{H(z, w) ; \quad(z, w) \in \mathbb{C} \times \mathbb{C}\} \subset M(2 \times 2 ; \mathcal{C}) $$ Show that $\mathcal{H}$ is a skew field, i.e. in $\mathcal{H}$ all the field axioms hold with the exception of the commutativity law for multiplication. Remark. The notation $\mathcal{H}$ is intended to remind us of Sir WuLLIAM ROWAN HAMILTON (1805-1865). One calls $\mathcal{H}$ the HAMILTON quaternions.

Hamilton's Quaternions (W. R. HAMLLTON, 1843)
We consider the following map
$$
\begin{array}{r}
H: \mathbb{C} \times \mathbb{C} \longrightarrow M(2 \times 2 ; \mathbb{C}) \\
(z, w) \mapsto H(z, w):=\left(\begin{array}{cc}
z & -w \\
\bar{w} & \bar{z}
\end{array}\right)
\end{array}
$$
and denote its image by
$$
\mathcal{H}:=\{H(z, w) ; \quad(z, w) \in \mathbb{C} \times \mathbb{C}\} \subset M(2 \times 2 ; \mathcal{C})
$$
Show that $\mathcal{H}$ is a skew field, i.e. in $\mathcal{H}$ all the field axioms hold with the exception of the commutativity law for multiplication.
Remark. The notation $\mathcal{H}$ is intended to remind us of Sir WuLLIAM ROWAN HAMILTON (1805-1865). One calls $\mathcal{H}$ the HAMILTON quaternions.

Key Concepts

Division Ring (Skew Field)

A division ring, also known as a skew field, is an algebraic structure in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative. Hamilton’s quaternions provide a quintessential example of a division ring since they satisfy all the field axioms except the commutativity of multiplication. This noncommutative property distinguishes division rings from commutative fields such as the real or complex numbers.

Matrix Representation

Matrix representations involve expressing elements of an abstract algebra as matrices, thereby allowing concrete computations and visualizations. In the case of Hamilton's quaternions, there exists a representation in terms of 2 × 2 complex matrices. This representation preserves the algebraic operations (addition and multiplication) and offers a bridge between abstract algebraic concepts and linear algebra techniques.

Quaternion Algebra

A quaternion algebra is a four-dimensional algebra over the real numbers (or another field) that generalizes the complex numbers. It consists of elements that can be expressed as a linear combination of one real basis element and three imaginary basis elements, and it exhibits a noncommutative multiplication structure. Hamilton's construction is the classical example of a quaternion algebra, and it is noted for its role in extending the concept of complex numbers to represent rotations in three dimensions.

Field Axioms

The field axioms include the properties of associativity, commutativity (for addition and, in the case of fields, multiplication), distributivity of multiplication over addition, existence of additive and multiplicative identities, and existence of additive inverses and multiplicative inverses for nonzero elements. In the context of Hamilton's quaternions, while all axioms hold true, the multiplication is noncommutative, which means they satisfy the axioms of a division ring rather than a commutative field.

Transcript

Please give Ace some feedback