Exercise 9.2

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}}

From now on we shall set $D = \Z[\omega]$ and $\lambda = 1 - \omega$. For $\mu$ in $D$ show that we can write $\mu = (-1)^a \omega^b \lambda^c\pi_1^{a_1}\pi_2^{a_2}\cdots\pi_t^{a_t}$, where $a,b,c$, and the $a_i$ are nonnegative integers and the $\pi_i$ are primary primes.

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}

Let $S$ the set containing $\lambda = 1 - \omega$ and all primary primes.

We show that
\begin{enumerate}
\item[(a)] every prime in $D$ is associate to a prime in $S$,
\item[(b)] no two primes in $S$ are associate.
\end{enumerate}

Let $\pi$ be a prime in $D$. There are three cases.

\begin{enumerate}
  \item[$\bullet$] If $
(\pi) = 3$, then $\pi$ is associate to $\lambda \in S$, thus $\pi \in \{1-\omega, -1+\omega, -2 - \omega, 2 + \omega, 1 + 2\omega, -1-2\omega\}$, and no associate of $\lambda$ is primary.
  \item[$\bullet$] If $
(\pi) = q^2$, where $q \equiv -1 \pmod 3, q>0,$ is a rational prime, then $\pi$ is associate to $q$ (Proposition 9.1.2), and $q$ is a primary prime.
  	The primes associate to $q$ are $q, -q, \omega q, -\omega q, -q - \omega q, q + \omega q$, so only $q$ is primary.
  \item[$\bullet$] If $
(\pi) = p$, where $p \equiv 1 \pmod 3$, then the proposition 9.1.4. shows that among the associates to $\pi$ exactly one is primary.
  \end{enumerate}
   Moreover, the norm of two primes belonging to two different cases are distinct, so two such primes are not associate.

By Theorem 3, Chapter 1, as $D = \Z[\omega]$ is a principal ideal domain,  every $\mu \in D$ is of the form
$$\mu = u \prod_{\pi \in S} \pi^{e(\pi)},$$
where $u$ is a unit, so $u = (-1)^a\omega^b$. Thus
$$\mu = (-1)^a \omega^b \lambda^c\pi_1^{a_1}\pi_2^{a_2}\cdots\pi_t^{a_t},$$ where the $\pi$ are primary primes, and $a,b,c$ and the $a_i$ are nonnegative integers.
\end{proof}

Exercise 9.2

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Exercise 9.2

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