Exercise 5.12

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}}

Let $f(x) \in \Z[x]$. We say that a prime $p$ divides $f(x)$ if there's an integer $n$ such that $p \mid f(n)$. Describe the prime divisors of $x^2 + 1$ and $x^2 -2$.

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}
$p$ divides $x^2+1$ iff there exists $a \in \Z$ such that $-1 \equiv a^2 \pmod p$, iff $p=2$ or $\legendre{-1}{p} = 1$ iff $ p=2$ or $p \equiv 1 \pmod 4$.

$p$ divides $x^2-2$ iff there exists $a \in \Z$ such that $2 \equiv a^2 \pmod p$, iff $p=2$ or  $\legendre{2}{p} = 1$ iff $p=2$ or $p \equiv \pm 1 \pmod 8$.
\end{proof}

Exercise 5.12

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Exercise 5.12

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