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
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 $$\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}$$.

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 $$3\in$$ $$\displaystyle{\mathbb{{{Z}}}}_{{2}}{\left[{x}\right]}$$
$$\displaystyle{{f}_{{1}}{\left({x}\right)}}={x}^{{3}}+{x}^{{2}}+{1}$$
$$\displaystyle{{f}_{{2}}{\left({x}\right)}}={x}^{{3}}+{x}+{1}$$