There's this claim which I want to prove. The claim goes as follows:
Prove that the polynomial $x^{2^n}+x^{2^{n-1}}+1$ is a product of at least $n$ irreducible polynomials.
I was thinking something along the line of $(x^2+x+1)(x^2-x+1)$ and then iterate it.
For $n=1$ it's $x^2+x+1$; for $x=2$ it's $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$.
So suppose that the assertion is correct for $n$ and prove it for $n+1$.
Now the induction step is where I am not sure, I know that $x^{2^n}+x^{2^{n-1}}+1$ by the induction hypothesis is a product of at least $n$ irreducible polynomials, but then how to show that $x^{2^{n+1}}+x^{2^n}+1=y^2+y+1$ where $y=x^{2^n}$ is a product of at least $n+1$ irreducible polynomials?
Use that $$ (x^{2^n}-x^{2^{n-1}}+1)(x^{2^n}+x^{2^{n-1}}+1)=x^{2^{n+1}}+x^{2^n}+1 $$ for the induction. In fact, $x^{2^n}-x^{2^{n-1}}+1$ is irreducible. So we obtain exactly $n$ irreducible factors.