We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q = xe + yI + zJ + tK \in \mathbb{H}$, $q^* = xe - yI - zJ - tK$ and $N(q) = x^2 + y^2 + z^2 + t^2$. We still assume that $p$ is an odd prime number. Justify that there exist $m \in \{1, \ldots, p-1\}$ and $\mu = xe + yI + zJ + tK \in \mathbb{G} \backslash \{0\}$ such that $N(\mu) = mp$. We choose $m$ minimal and assume that $m > 1$. a) Show that if $m$ were even, an even number of the integers $x, y, z, t$ would be odd and lead to a contradiction. You may write $\left(\frac{x-y}{2}\right)^2 + \left(\frac{x+y}{2}\right)^2 = \frac{x^2 + y^2}{2}$. b) We assume $m$ is odd. Show that there exists $\nu \in \mathbb{G}$ such that $N(\mu - m\nu) < m^2$. c) Prove that $\mu' = \frac{1}{m}\mu(\mu - m\nu)^*$ is in $\mathbb{G} \backslash \{0\}$ and that $N(\mu')$ is a multiple of $p$ strictly less than $mp$. Conclude.
We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q = xe + yI + zJ + tK \in \mathbb{H}$, $q^* = xe - yI - zJ - tK$ and $N(q) = x^2 + y^2 + z^2 + t^2$.\\
We still assume that $p$ is an odd prime number. Justify that there exist $m \in \{1, \ldots, p-1\}$ and $\mu = xe + yI + zJ + tK \in \mathbb{G} \backslash \{0\}$ such that $N(\mu) = mp$. We choose $m$ minimal and assume that $m > 1$.\\
a) Show that if $m$ were even, an even number of the integers $x, y, z, t$ would be odd and lead to a contradiction.
You may write $\left(\frac{x-y}{2}\right)^2 + \left(\frac{x+y}{2}\right)^2 = \frac{x^2 + y^2}{2}$.\\
b) We assume $m$ is odd. Show that there exists $\nu \in \mathbb{G}$ such that $N(\mu - m\nu) < m^2$.\\
c) Prove that $\mu' = \frac{1}{m}\mu(\mu - m\nu)^*$ is in $\mathbb{G} \backslash \{0\}$ and that $N(\mu')$ is a multiple of $p$ strictly less than $mp$. Conclude.