Is there a systematic way to find irreducible polynomials?

Question

I am preparing for an exam and I'm looking for a systematic way to find all irreducible polynomials. I am given the following problem > Find all irreducible polynomials of degree 3 with coefficients over Z2. I know that a polynomial is reducible if it can be represented as a product of two or more polynomials, but I can't find an efficient algorithm which can help me to list all the irreducible polynomials. Any suggestions?

Answer 1

If $f(x)\in \mathbb Z_2[x]$ and has degree $3$,then $f(x)=x^3+ax^2+bx+c$ and we have the possible states:

$a=0,b=0,c=0 \Rightarrow f(x)=x^3$

$a=1,b=0,c=0 \Rightarrow f(x)=x^3+x^2$

$a=1,b=1,c=0 \Rightarrow f(x)=x^3+x^2+x$

$a=1,b=1,c=1 \Rightarrow f(x)=x^3+x^2+x+1$

$a=1,b=0,c=1 \Rightarrow f(x)=x^3+x^2+1$

$a=0,b=1,c=0 \Rightarrow f(x)=x^3+x$

$a=0,b=1,c=1 \Rightarrow f(x)=x^3+x+1$

$a=0,b=0,c=1 \Rightarrow f(x)=x^3+1$

$f(x)$ is irreducible polynomial in $\mathbb Z_2[x]$ if $0$ or $1$ isnot roots, then $f(x)=x^3+x^2+1$ and $f(x)=x^3+x+1$ are irreducible polynomials in $\mathbb Z_2[x]$.