Exercise 1.38

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Suppose that $\pi \in \Z[i]$ and that $\lambda(\pi) = p$ is a prime in $\Z$. Show that $\pi$ is a prime in $\Z[i]$. Show that the corresponding result holds in $\Z[\omega]$ and $\Z[\sqrt{-2}]$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof} If $\pi = \alpha \beta$, where $\alpha, \beta \in \Z[i]$, then $p = \lambda(\pi) = \lambda(\alpha) \lambda(\beta)$. As $p$ is a prime in $\Z$, and $\lambda(\alpha) \geq 0,,\lambda(\beta) \geq 0$, $\lambda(\alpha) = 1$ or $\lambda(\beta)=1$, so (Ex.1.33) $\alpha$ or $\beta$ is an unit. Consequently, $\pi$ is irreducible in $\Z[i]$. As $\Z[i]$ is a PID,  $\pi$ is a prime in $\Z[i]$ (Prop. 1.3.2 Corollary 2).

As $\Z[\omega]$ and $\Z[\sqrt{-2}]$ are Euclidean domains, the same result is true in these principal ideals domains.
\end{proof}

Exercise 1.38

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Exercise 1.38

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