Question

List all of the polynomials of degrees 2 and 3 in \mathbb{Z}_2[x]. Find all of the irreducible polynomials of degrees 2 and 3 in \mathbb{Z}_2[x].

Polynomials
ANSWERED
asked 2021-09-18
List all of the polynomials of degrees 2 and 3 in \(\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\). Find all of the irreducible polynomials of degrees 2 and 3 in \(\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\).

Expert Answers (1)

2021-09-19

Let \(\displaystyle{a}{x}^{{3}}+{b}{x}^{{2}}+{c}{x}+{d}\in{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\) be a polynomial of degree 3.
then we must have a=1
for this polynomial to be irreduicble we must also d=1
since otherwise we will have a polynomial \(\displaystyle{x}^{{3}}+{b}{x}^{{2}}+{c}{x}={x}{\left({x}^{{2}}+{b}{x}+{c}\right)}\)
that can be factored an therefore reducible.
so the polynomial \(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}+{b}{x}^{{2}}+{c}{x}+{1}\), where b, \(\displaystyle{c}\in{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\)
so the polynomial has no zero in \(\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\)
so, we may not have \(f(0)=0\) of \(f(1)=0\)
now f(0)=1 no matter chosen \(f(1)=b+c\)
hence we have to choose b and c so, that \(b+c=1\)
and the two possibilities \(\displaystyle{b}={1},{c}={0}{\quad\text{or}\quad}{b}={0},{c}={1}\).
that there are two irreducible polynomials of degree \(3\in\) \(\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}\)
\(\displaystyle{{f}_{{1}}{\left({x}\right)}}={x}^{{3}}+{x}^{{2}}+{1}\)
\(\displaystyle{{f}_{{2}}{\left({x}\right)}}={x}^{{3}}+{x}+{1}\)

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