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 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