Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.14
Solution: Define $\varphi : \mathbb{H} \rightarrow M_4(\mathbb{R})$ as follows. $$a+bi+cj+dk \mapsto \begin{bmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\ -d & -c & b & a \end{bmatrix}$$We will show that this mapping is an injective ring homomorphism. To that end, let $\alpha = a_1 + b_1i + c_1j + d_1k$ and $\beta = a_2 + b_2i + c_2j + d_2k$. Then we have the following.\begin{align*}\varphi(\alpha + \beta) =&\ \varphi((a_1 + b_1i + c_1j + d_1k)+(a_2 + b_2i + c_2j + d_2k))\\
=&\ \varphi((a_1 + a_2) + (b_1+b_2)i + (c_1+c_2)j + (d_1+d_2)k)\\
=&\ \begin{bmatrix} a_1+a_2 & b_1+b_2 & c_1+c_2 & d_1+d_2 \\ -b_1-b_2 & a_1+a_2 & -d_1-d_2 & c_1+c_2 \\ -c_1-c_2 & d_1+d_2 & a_1+a_2 & -b_1-b_2 \\ -d_1-d_2 & -c_1-c_2 & b_1+b_2 & a_1+a_2 \end{bmatrix}\\
=&\ \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ -b_1 & a_1 & -d_1 & c_1 \\ -c_1 & d_1 & a_1 & -b_1 \\ -d_1 & -c_1 & b_1 & a_1 \end{bmatrix} + \begin{bmatrix} a_2 & b_2 & c_2 & d_2 \\ -b_2 & a_2 & -d_2 & c_2 \\ -c_2 & d_2 & a_2 & -b_2 \\ -d_2 & -c_2 & b_2 & a_2 \end{bmatrix}\\
=&\ \varphi(a_1 + b_1i + c_1j + d_1k) + \varphi(a_2 + b_2i + c_2j + d_2k)\\
=&\ \varphi(\alpha) + \varphi(\beta)\end{align*}\begin{align*}\varphi(\alpha\beta) =&\ \varphi((a_1 + b_1i + c_1j + d_1k)(a_2 + b_2i + c_2j + d_2k))\\
=&\ \varphi((a_1a_2 – b_1b_2 – c_1c_2 – d_1d_2) + (a_1b_2 + b_1a_2 + c_1d_2 – d_1c_2)i + (a_1c_2 – b_1d_2 + c_1a_2 + d_1b_2)j + (a_1d_2 + b_1c_2 – c_1b_2 + d_1a_2)k)\\
=&\ \begin{bmatrix} a_1a_2 – b_1b_2 – c_1c_2 – d_1d_2 & a_1b_2 + b_1a_2 + c_1d_2 – d_1c_2 & a_1c_2 – b_1d_2 + c_1a_2 + d_1b_2 & a_1d_2 + b_1c_2 – c_1b_2 + d_1a_2 \\ -a_1b_2 – b_1a_2 – c_1d_2 + d_1c_2 & a_1a_2 – b_1b_2 – c_1c_2 – d_1d_2 & -a_1d_2 – b_1c_2 + c_1b_2 – d_1a_2 & a_1c_2 – b_1d_2 + c_1a_2 + d_1b_2 \\ -a_1c_2 + b_1d_2 – c_1a_2 – d_1b_2 & a_1d_2 + b_1c_2 – c_1b_2 + d_1a_2 & a_1a_2 – b_1b_2 – c_1c_2 – d_1d_2 & -a_1b_2 – b_1a_2 – c_1d_2 + d_1c_2 \\ -a_1d_2 – b_1c_2 + c_1b_2 – d_1a_2 & -a_1c_2 + b_1d_2 – c_1a_2 – d_1b_2 & a_1b_2 + b_1a_2 + c_1d_2 – d_1c_2 & a_1a_2 – b_1b_2 – c_1c_2 – d_1d_2 \end{bmatrix}\\
=&\ \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ -b_1 & a_1 & -d_1 & c_1 \\ -c_1 & d_1 & a_1 & -b_1 \\ -d_1 & -c_1 & b_1 & a_1 \end{bmatrix} \cdot \begin{bmatrix} a_2 & b_2 & c_2 & d_2 \\ -b_2 & a_2 & -d_2 & c_2 \\ -c_2 & d_2 & a_2 & -b_2 \\ -d_2 & -c_2 & b_2 & a_2 \end{bmatrix}\\
=&\ \varphi(a_1 + b_1i + c_1j + d_1k) \cdot \varphi(a_2 + b_2i + c_2j + d_2k)\\
=&\ \varphi(\alpha) \cdot \varphi(\beta)\end{align*}Thus $\varphi$ is a ring homomorphism.
Suppose now that $\alpha = a+bi+cj+dk \in \mathsf{ker}\ \varphi$; clearly, then, we have $a = b = c= d = 0$, so that $\alpha = 0$. Thus $\varphi$ is injective.
By Proposition 5 in the text, $\mathsf{im}\ \varphi$ is a subring of $M_4(\mathbb{R})$ which is isomorphic to $\mathbb{H}$.
