Exercise 8.14

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 $p\equiv 1 \pmod n$ and that $\chi$ is a character of order $n$. Show that $g(\chi)^n \in\Z[\zeta]$, where $\zeta = e^{2\pi i/n}$.

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}
From Proposition 8.3.3 we know that
$$g(\chi)^n = \chi(-1) p J(\chi,\chi) J(\chi, \chi^2) \cdots J(\chi, \chi^{n-2}).$$
Let $\mathbb{U}_n =\{x \in \C\ | \ x^n = 1\} = \{1, \zeta, \ldots, \zeta^{n-1}\}$, with $\zeta = e^{2\pi i/n}$, the group of $n$-th roots of unity. As the order of $\chi$ is $n$, for all $x \in \F_p^*$, $(\chi(x))^n = \chi^n(x) = \varepsilon(x) = 1$, so $\chi(x) \in \mathbb{U}_n$, and also $\chi^k(x) = (\chi(x))^k$.

Therefore $J(\chi,\chi^k) = \sum\limits_{x+y=1} \chi(x) \chi^k(y) \in \Z[\zeta]$. Moreover $\chi(-1) = \pm 1$, so $\chi(-1)$ and $p$ are in $\Z[\zeta]$. In conclusion $g(\chi)^n \in \Z[\zeta]$.
\end{proof}

Exercise 8.14

Answers

Comments

Exercise 8.14

Answers

Comments

Add answer