Find a polynomial of degree 5 that is irreducible over $\mathbb{Z}_3$.

Question

Can someone please let me know if this looks ok? Thanks in advance!

An irreducible degree $5$ polynomial over $\mathbb{Z}_3$is one such that $f(0)\neq0,f(1)\neq0,f(2)\neq0$.

Take e.g. $p(x)=x^5+x^4+x^3+x^2+1$

$p(0)=1$

$p(1)=2$

$p(2)=1$

And, $$x^2,x^2+1, x^2+x+1\ \nmid\ x^5+x^4+x^3+x^2+1.$$

$\therefore x^5+x^4+x^3+x^2+1$ is an irreducible polynomial over $\mathbb{Z}_3$.

Answer 1

Since it has degree $5$, it would need to have a root ( but it doesn't), or be divisible by an irreducible quadratic.

But, did you check $x^2+2x+2$?

There may be others.

Answer 2

Your problem is to find an irreducible polynomial of degree $5$ over $\mathbb Z_3$. You haven't found one yet, because $$x^5+x^4+x^3+x^2+1=(x^2+x+2)(x^3+2x+2).$$

Note that the irreducible (monic) polynomials of degree $2$ are $x^2+1$ and $x^2+x+2$ and $x^2+2x+2$. You need to check all three of them as possible factors.

Hint. Try $x^5+2x+1$.

Answer 3

You can also use this proposition from Dummit and Foote, and Sage to get these polynomials.

Proposition 18. The polynomial $x^{p^{n}}-x$ is precisely the product of all the distinct irreducible polynomials in $\mathbb{F}_{p}[x]$ of degree $d$ where $d$ runs through all divisors of $n$.

So $x^{5^{3}}-x=x^{243}-x$ is the product of all the distinct irreducible polynomials in $\mathbb{F}_{3}[x]$ of degree $1$ or $5$. Using Sage:

F=GF(3) 
x=F['x'].0 
factor(x^243-x)

We get

x * (x + 1) * (x + 2) * (x^5 + 2*x + 1) * (x^5 + 2*x + 2) * (x^5 + x^2 + x + 2) * (x^5 + 2*x^2 + x + 1) * (x^5 + x^3 + x + 1) * (x^5 + x^3 + x + 2) * (x^5 + x^3 + x^2 + 2) * (x^5 + x^3 + x^2 + 2*x + 2) * (x^5 + x^3 + 2*x^2 + 1) * (x^5 + x^3 + 2*x^2 + 2*x + 1) * (x^5 + 2*x^3 + x^2 + 1) * (x^5 + 2*x^3 + x^2 + x + 2) * (x^5 + 2*x^3 + x^2 + 2*x + 2) * (x^5 + 2*x^3 + 2*x^2 + 2) * (x^5 + 2*x^3 + 2*x^2 + x + 1) * (x^5 + 2*x^3 + 2*x^2 + 2*x + 1) * (x^5 + x^4 + 2) * (x^5 + x^4 + x + 2) * (x^5 + x^4 + 2*x + 1) * (x^5 + x^4 + x^2 + 1) * (x^5 + x^4 + x^2 + x + 1) * (x^5 + x^4 + x^2 + 2*x + 2) * (x^5 + x^4 + x^3 + x + 1) * (x^5 + x^4 + x^3 + x^2 + 2*x + 1) * (x^5 + x^4 + x^3 + 2*x^2 + x + 1) * (x^5 + x^4 + x^3 + 2*x^2 + x + 2) * (x^5 + x^4 + 2*x^3 + 1) * (x^5 + x^4 + 2*x^3 + 2*x + 2) * (x^5 + x^4 + 2*x^3 + x^2 + 2) * (x^5 + x^4 + 2*x^3 + x^2 + x + 1) * (x^5 + x^4 + 2*x^3 + 2*x^2 + 1) * (x^5 + x^4 + 2*x^3 + 2*x^2 + 2) * (x^5 + 2*x^4 + 1) * (x^5 + 2*x^4 + x + 1) * (x^5 + 2*x^4 + 2*x + 2) * (x^5 + 2*x^4 + 2*x^2 + 2) * (x^5 + 2*x^4 + 2*x^2 + x + 2) * (x^5 + 2*x^4 + 2*x^2 + 2*x + 1) * (x^5 + 2*x^4 + x^3 + x + 2) * (x^5 + 2*x^4 + x^3 + x^2 + x + 1) * (x^5 + 2*x^4 + x^3 + x^2 + x + 2) * (x^5 + 2*x^4 + x^3 + 2*x^2 + 2*x + 2) * (x^5 + 2*x^4 + 2*x^3 + 2) * (x^5 + 2*x^4 + 2*x^3 + 2*x + 1) * (x^5 + 2*x^4 + 2*x^3 + x^2 + 1) * (x^5 + 2*x^4 + 2*x^3 + x^2 + 2) * (x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 1) * (x^5 + 2*x^4 + 2*x^3 + 2*x^2 + x + 2)