# Quaternions form Skew Field

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## Theorem

The set $\H$ of quaternions forms a skew field under the operations of addition and multiplication.

## Proof

From Ring of Quaternions is Ring we have that $\H$ forms a ring.

From Multiplicative Identity for Quaternionsâ€Ž we have that $\mathbf 1$ is the identity for quaternion multiplication.

From Multiplicative Inverse of Quaternion we have that every element of $\H$ except $\mathbf 0$ has an inverse under quaternion multiplication.

So $\H \setminus \set 0 = \H^*$ is a group.

Hence $\H$ forms a division ring.

But quaternion multiplication is specifically not commutative, for example:

- $\mathbf i \mathbf j = \mathbf k, \ \mathbf j \mathbf i = -\mathbf k$

So $\H$ forms a skew field under addition and multiplication.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$ - 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(3)$