Number of factors of a polynomial in F_2

[cyclic]

Homework Statement

Prove that the polynomial 𝑥^(6y)+x^(5y)+x^(4y)+𝑥^(3y)+1 always has four or more factors in 𝔽_2 if 𝑦 is not a power of 3.

Relevant Equations

Perhaps the fact that 𝑥^(2^𝑛)−𝑥 is the product of all monic primes in 𝔽_2[𝑥]
of degree d | n may be of help here, but I'm not sure. I would appreciate any help/guidance here.

This is a pattern I noticed when playing around with Mathematica. Is there any way to rigorously prove this? I was not able to find any literature concerning the number of factors in a finite field, especially because this is called a "pentanomial" in said literatures. These don't have much theory behind them, but since this polynomial looked nice in terms of the degrees of its exponents, there should be an easier way somehow.

 

[fresh_42]

I would start by setting $z=x^y$ because the $y$ disturbs the most. We thus have to examine
\begin{align*}
z^6+z^3+z^2+1&=(z+1)^6+z^4+z^3=(z+1)\left[(z+1)^5+z^3\right]\\
\end{align*}
I'm not sure whether this helps, but it is now a degree less to factor.

 

[cyclic]

I would start by setting $z=x^y$ because the $y$ disturbs the most. We thus have to examine
\begin{align*}
z^6+z^3+z^2+1&=(z+1)^6+z^4+z^3=(z+1)\left[(z+1)^5+z^3\right]\\
\end{align*}
I'm not sure whether this helps, but it is now a degree less to factor.

Apologies, I realized that I typed the question wrong. Thank you for the response, though. I've edited the question. Looking at the factorizations in Mathematica, I'm not sure that a general factorization exists, but I may be wrong.

 

[fresh_42]

Apologies, I realized that I typed the question wrong.

Your modified polynomial results in
$$
z^6+z^5+z^4+z^3+1=(z^2 + z + 1) (z^4 + z + 1) \pmod{2} \text{ with }z=x^y
$$
This reduces the problem to two polynomials of lower degree. A check with $y=3^0=1$ and $y=2$ resulted in
\begin{align*}
x^6+x^5+x^4+x^3+1&=(x^2 + x + 1) (x^4 + x + 1) \pmod{2}\\
x^{12}+x^{10}+x^6+1&=(x^2 + x + 1)^2 (x^4 + x + 1)^2 \pmod{2}
\end{align*}
Maybe this is the pattern.