Is $ \mathbb{F}_p^{*} $ an integral ring

Question

In my textbook $p$ is a prime numberand $\mathbb{F}_p^*$ is cyclic.

There is $x\in \mathbb{F}_p^*$ such that $x^5=-1$ which means that $(x+1)(x^4-x^3+x^2-x+1)=0$

The conclusion is that $(x^4-x^3+x^2-x+1)=0$ but I don't understand why?

Is $\mathbb{F}_p^*$ an integral ring?

From wikipedia I see that $\mathbb{F}_n^*$ is an integral ring implies $n$ is prime but not the opposite.

Many thanks for your help.

Answer 1

When you are asking 'Is $\mathbb{F}_p^{*}$ an integral ring?' you should first ask yourself 'Is it a ring?'. If you think it is then what is addition, what is multiplication?

Hint: Observe the identity of abelian group.