Suppose that f(x) is a fifth-degree irreducible polynomial over Z2. Prove that x is a generator of the cyclic group (Z2[x]/)*

Question

Suppose that f(x) is a fifth-degree irreducible polynomial over Z2. Prove that x is a generator of the cyclic group (Z2[x]/<f(x)>)*

helder_edwin · Answer

Since $f(x)$ is irreducible, then $J=\langle f(x)\rangle$ is maximal. Therefore $\mathbb{Z}_2[x]/J$ is a field.

helder_edwin · Answer

Let $H=\mathbb{Z}_2[x]/J$. The elements of $H$ have the form
$$\large [ax^4+bx^3+cx^2+dx+e] $$
where $a,b,c,d,e\in\mathbb{Z}_2$. Then $H$ has $2^5=32$ elements.

helder_edwin · Answer

I have a notation question: does $H^*$ stand for the isomorphic copy of $\mathbb{Z}_2$ in $H$? Or, does it stand for the non-zero elements of $H$?

anonymous · Answer

I'm not sure. Our professor never used the * notation before in class, but then gave us homework questions with it on there