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## Embed the Hamiltonian quaternions in a ring of real matrices

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 &…

## A norm on the integral Hamiltonians

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.25 Let $I$ be the ring of integral Hamiltonian Quaternions and define $N : I \rightarrow \mathbb{Z}$…

## Compute the center of the Hamiltonian quaternions

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.8 Describe $Z(\mathbb{H}), where \mathbb{H}$ denotes the Hamiltonian Quaternions. Prove that \$\{a+bi \ |\ a,b \in \mathbb{R}…