Show that a polynomial over $\mathbb{Z}_{2}$ is irreducible Given the polynomial: $p(x)=x^4+x^3+x^2+x+1$ over $\mathbb{Z}_{2}$, to show that it is irreducable, is it enough to show that $p(0)=p(1)=1$?

On factorising into quadratics, just try by hand. The coefficient of $x^2$ has to be $1$ in both factors to give $x^4$ and the constant in each factor has to be $1$ too to give a constant term $1$ in the product.
So $$(x^2+ax+1)(x^2+bx+1)=x^4+(a+b)x^3+abx^2+(a+b)x+1=x^4+x^3+x^2+x+1$$
We have $a+b=1$ so $a$ and $b$ have different parities, and one is $0$. But then $ab=0$ and the coefficient of $x^2$ is zero. So there is no possible factorisation.
