Quaternions form Skew Field

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 Quaternions‎ 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): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
• 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: $(3)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): division ring