We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$. a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication. b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$. c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?
We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q \in \mathbb{H}$, $N(q) = x^2 + y^2 + z^2 + t^2$ where $q = xe + yI + zJ + tK$.\\
a) Show that $\mathbb{G}$ is a subgroup of $\mathbb{H}$ for addition and that it is stable under multiplication.\\
b) Show that for every $q \in \mathbb{H}$, there exists $\mu \in \mathbb{G}$ such that $N(q - \mu) \leqslant 1$.\\
c) What is the set of $q \in \mathbb{H}$ such that $\forall \mu \in \mathbb{G}, N(q - \mu) \geqslant 1$?