Problem 4

Let $p(x)\in\mathbb{Z}_{2}$ be a polynomial of degree 7 that is irreducible in $\mathbb{Z}_{2}$. Prove or disprove the following statements.

1. $\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle$ is a field.

2. $\left(\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle\right)^{\ast}$ is a group under coset multiplication.

3. Every non-identity element in $\left(\mathbb{Z}_{2}[x]/\left\langle p(x)\right\rangle\right)^{\ast}$ is a generator.