Problem 7

We show a failure case of the Mod $p$ Irreducibility Test (1.18). Let the polynomial $f(x)=x^{4}+1$.

1. Prove that $f(x)$ is irreducible over $\mathbb{Q}$.

2. Show that $f(x)$ is reducible over $\mathbb{Z}_{2}$ via factorization.For the rest of this part, assume $p>2$.

   1. Suppose there is an $r\in\mathbb{Z}_{p}$ satisfying $r^{2}=k$ for some $k$ (to be specified). Deduce a factorization of $f(x)$ in the following cases.

      1. $k=2$

      2. $k=-1$

      3. $k=-2$

   2. Let $\mathbb{Z}_{p}^{\ast}=\mathbb{Z}_{p}\setminus\{0\}$. Explain why $\mathbb{Z}_{p}^{\ast}$ is a group under multiplication modulo $p$. In particular, explain why $\mathbb{Z}_{p}^{\ast}=\mathcal{U}_{p}$, the group of units modulo $p$.

   3. Explain why $\mathcal{U}_{p}$ is a cyclic group of even order.

   4. Hence, prove that one of the cases in (b)(i) must occur.