Exercise 2.2.1 (First examples, part 1)

Question

If $f(x) \equiv 0 \pmod p$ has exactly $j$ solutions with $p$ a prime, and $g(x) \equiv 0 \pmod p$ has no solution, prove that $f(x) g(x) \equiv 0 \pmod p$ has exactly $j$ solutions.

Richard Ganaye · Accepted Answer

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

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

\begin{proof}
Since $p$ is a prime number,
\begin{align*}
f(x) g(x) \equiv 0 \pmod p &\iff f(x) \equiv 0 \pmod p 	ext{ or } g(x) \equiv 0 \pmod p\
&\equiv  f(x) \equiv 0 \pmod p,
\end{align*}
because $g(x) \equiv 0 \pmod p$ is always false.

So, there are as many solutions of $f(x) g(x) \equiv 0 \pmod p$ than of $f(x) \equiv 0 \pmod p$.
\end{proof}

Exercise 2.2.1 (First examples, part 1)

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Exercise 2.2.1 (First examples, part 1)

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