Exploring Irreducible Polynomial and Reducing Techniques for g(x)

[ElDavidas]

Hi, I'm trying to show whether the polynomial

g(x) = x^8+ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

is irreducible or not.

So far I have evaluated g(x+1) and applied Eisenstein's theorem to it. From what I gather it doesn't appear to be irreducible. Is this right, because I reckon it should be irreducible? This may just be a simple calculation error.

And if g(x) is reducible, how do I go about reducing the polynomial more?

Thanks

 

[AKG]

What is g(x)(x-1), and what are its roots?

 

[ElDavidas]

Of course, that would be helpful.

g(x)(x-1) = (x^9-1)

and the roots are \alpha existing in the complex numbers such that
\alpha^9 = 1

 

[AKG]

So you can solve your problem now right?

 

[robert Ihnot]

Two of the roots are cos(120)+isin(120)=w (cube root of 1,) and cos(240)+isin(240)=w^2. Combining these two roots (x-w)(x-w^2)=x^2+x+1.

This then divides the polynominal giving: (1+x+x^2)(x^6+x^3+1).