Exercise 7.16

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}}

Calculate the monic irreducible polynomials of degree 4 in $\Z/2\Z[x]$.

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}
Write $F_d$ the product of irreducible monic polynomials in $\mathbb{F}_2[x]$.

Theorem 2 gives $$x^{16}-x=x^{2^4}-x = \prod\limits_{d\mid 4} F_d(x) = F_1(x) F_2(x) F_4(x)$$ 
and

$$x^{4}-x=x^{2^2}-x = \prod\limits_{d\mid 2} F_d(x) = F_1(x) F_2(x)$$

so $F_4(x) = \frac{x^{16}-x}{x^4-x} = \frac{x^{15}-1}{x^3-1} = x^{12}+x^9+x^6+x^3+1$,

$F_4(x) = (x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1)$.

Among the 16 monic polynomials of degree 4 in $\mathbb{F}_2[x]$, 3 are irreducible :
\begin{align*}
P_1(x) &= x^4+x^3+x^2+x+1,\
 P_2(x)&=x^4+x+1,\
  P_3(x)&=x^4+x^3+1.
\end{align*}

With sage : 
\begin{verbatim}
sage: A = PolynomialRing(GF(2),'x')
sage: x = A.gen()
sage: f = (x^16-x)/(x^4-x)
sage: factor(f)
(x^4 + x + 1) * (x^4 + x^3 + 1) * (x^4 + x^3 + x^2 + x + 1)
\end{verbatim}
\end{proof}

Exercise 7.16

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Exercise 7.16

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