Exercise 8.2

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}}

extbf{(false sentence)} With the notation of Exercise 1 show that $N(x^m = a) = N(x^d = a)$ and conclude that if $d_i = (m_i,p-1)$, then $\sum_i a_ix^{m_i} = b$ and $\sum_i a_i x^{d_i} = b$ have the same number of solutions.

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}}

\bigskip

This result is false. I give a counterexample with $p=5$ :  $x + x^3 =0 \in \F_5[x]$ has 3 solutions $0,2,-2$.  As $3\wedge (p-1) = 3 \wedge 4 = 1$, the reduced equation is $x + x = 0$, which has an unique solution $0$. The true sentence is :

\bigskip	extbf{Ex. 8.2}
{\it With the notation of Exercise 1 show that $N(x^m = a) = N(x^d = a)$ and conclude that if $d_i = (m_i,p-1)$, then $\sum_i a_ix_i^{m_i} = b$ and $\sum_i a_i x_i^{d_i} = b$ have the same number of solutions.
}

\begin{proof}
From Ex. 8.1, we know that
$$N(x^m=a) = \sum_{\chi^d=\varepsilon} \chi(a) = N(x^d=a).$$
Using this result, we obtain
\begin{align*}
N\left(\sum\limits_{i=1}^l a_i x_i^{m_i} = bight) &= \sum\limits_{a_1u_1+\cdots+a_l u_l= b}\, \prod\limits_{i=1}^l N(x^{m_i} = u_i)\
&= \sum\limits_{a_1u_1+\cdots+a_l u_l= b}\, \prod\limits_{i=1}^l N(x^{d_i} = u_i)\
&=N\left(\sum\limits_{i=1}^l a_i x_i^{d_i} = bight).
\end{align*}
\end{proof}

Exercise 8.2

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Exercise 8.2

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