Let θ=((2 πi)/(5)). We write ℤ[θ]={a0+a1 θ+a2 θ^2+a3 θ^3: a0, a1, a2, a3 ∈ℤ} . Show that ℤ[θ] is

Question

Accepted Answer

Step 1:u000ALetu0027s identify what $u005Ctheta$ represents. We have $u005Ctheta u003D u005Cexpu005Cleft(u005Cfrac{2u005Cpi i}{5}u005Cri

Let $\theta=\exp \left(\frac{2 \pi i}{5}\right)$. We write $$ \mathbb{Z}[\theta]=\left\{a_0+a_1 \theta+a_2 \theta^2+a_3 \theta^3: a_0, a_1, a_2, a_3 \in \mathbb{Z}\right\} . $$ Show that $\mathbb{Z}[\theta]$ is a PID.