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Contents Dark Mode Prev Up Next \( \renewcommand{\Re}{\operatorname{Re}}
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Section 2.4 Hamilton quaternion rings
Specimen 26 . Quaternion rings.
Let \(R\) be a nontrivial commutative ring, and assume \(1+1\ne 0\) in \(R\text{.}\) We define \(H(R)\) to be the set of all formal sums
\begin{gather*}
a_1+a_2i+a_3j+a_4k\text{,}
\end{gather*}
where \(a_i\in R\) for all \(1\leq i\leq 4\text{.}\) We define addition on \(H(R)\) as
\begin{gather*}
(a_1+a_2i+a_3j+a_4k)+(b_1+b_2i+b_3j+b_4k) \\
= (a_1+b_1)+(a_2+b_2)i+(a_3+b_3)j+(a_4+b_4)k\text{.}
\end{gather*}
We define multiplication to be the unique binary operation \(\cdot\) that distributes over the addition operation \(+\text{,}\) satisfies
\begin{align}
b(a_1+a_2i+a_3j+a_4k) \amp =(a_1+a_2i+a_3j+a_4k)b \tag{2.4.1}\\
\amp =ba_1+ba_2i+ba_3j+ba_4k\tag{2.4.2}
\end{align}
for all
\begin{align*}
b \amp =b+0i+0j+0k\in H(R)\text{,}
\end{align*}
and satisfies
\begin{align}
i^2\amp =j^2=k^2=-1 \tag{2.4.3}\\
ij \amp = -ji= k\tag{2.4.4}\\
jk \amp = -kj= i\tag{2.4.5}\\
ki \amp = -ik= j\text{.}\tag{2.4.6}
\end{align}
The triple \((H(R), +, \cdot)\) is a ring called the Hamilton quaternion ring over (or with coefficients in) \(R\) .
In the special case where
\(R=\R\text{,}\) we denote
\(\HH=H(\R)\) and call
\(\HH\) simply the
Hamilton quaternions .
Definition 2.4.1 . Quaternion conjugation and norm.
Let \(R\) be a nontrivial commutative ring. Given \(q=a_1+a_2i+a_3j+a_4k\in H(R)\text{,}\) its quaternion conjugate , denoted \(\overline{q}\) is defined as
\begin{align}
\overline{q} \amp =a_1-a_2i-a_3j-a_4k\text{.}\tag{2.4.7}
\end{align}
The quaternion norm is the function \(N\colon H(R)\rightarrow R\) defined as
\begin{gather}
N(q)=\overline{q}q\text{.}\tag{2.4.8}
\end{gather}
A straightforward computation shows that if \(q=a_1+a_2i+a_3j+a_4k\text{,}\) then
\begin{align*}
N(q) \amp =a_1^2+a_2^2+a_3^2+a_4^2\text{.}
\end{align*}
Theorem 2.4.2 . Quaternion conjugation and norm.
Let \(R\) be a nontrivial commutative ring.
\(\overline{aq+bq'}=a\overline{q}+b\overline{q'}\) for all
\(a,b\in R\) and
\(q,q'\in H(R)\text{.}\)
\(\overline{qq'}=\overline{q'}\, \overline{q}\) for all
\(q,q'\in H(R)\text{.}\)
\(N(qq')=N(q)N(q')\) for all
\(q,q'\in H(R)\text{.}\)
The following statements are equivalent.
\(q\in (H(R))^*\text{.}\)
\(\overline{q}\in (H(R))^*\text{.}\)
Proof. This will be given as a homework exercise.
Corollary 2.4.3 . \(\HH\) is a division ring.
The Hamilton quaternions
\(\HH=H(\R)\) is a division ring.
Proof.
For \(q=a_1+a_2i+a_3j+a_4k\in \HH\text{,}\) we have
\begin{align}
q\ne 0 \amp \iff a_i\ne 0 \text{ for some } i \tag{2.4.9}\\
\amp \iff a_1^2+a_2^2+a_3^2+a_4^2\ne 0 \amp (a_i^2> 0 \iff a_i\ne 0)\tag{2.4.10}\\
\amp \iff N(q)\ne 0\tag{2.4.11}\\
\amp \iff N(q)\in \R^*\tag{2.4.12}\\
\amp \iff q\in \HH^*\text{.}\tag{2.4.13}
\end{align}
Thus \(\HH^*=\HH-\{0\}\text{,}\) as desired.
