Exercise 2.3.15 (Solve $x^3 + 4x + 8 \equiv 0 \pmod{15}$)

Question

Solve the congruence $x^3+4x + 8 \equiv 0 \pmod {15}$.

Richard Ganaye · Accepted Answer

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

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

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

\begin{proof} Let $f(x) = x^3 + 4x + 8$. Reducing modulo $5$, we obtain
$$f(-2) \equiv 2, \quad f(-1) \equiv 3,\quad  f(0) \equiv 3,\quad  f(1) \equiv 3, \quad f(2) \equiv 4 \pmod 5.$$
Since $\{-2,-1,0,1,2\}$ is a complete residue system modulo $5$, the congruence $f(x) \equiv 0 \pmod 5$ has no solution in $\Z$.

A fortiori, the congruence $f(x) \equiv 0 \pmod {15}$ has no solution in $\Z$.
\end{proof}

Check with Sage:
\begin{verbatim}
sage: R.<x> = GF(5)[]
sage:  f = x^3 + 4*x + 8
sage: f.is_irreducible()
True
\end{verbatim}
A fortiori, $x^3 + 4x + 8$ has no solution in $\F_5 = \mathrm{GF}(5)$.

Exercise 2.3.15 (Solve $x^3 + 4x + 8 \equiv 0 \pmod{15}$)

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Exercise 2.3.15 (Solve $x^3 + 4x + 8 \equiv 0 \pmod{15}$)

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