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]/)*

Answer 1

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

Answer 2

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.

Answer 3

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 \(H\)?

Answer 4

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