Polynomial divibility over finite field

Question

Let $p$ be a prime number, and let F := $\mathbb{Z}/p\mathbb{Z}$ and let $f(t) \in F[t]$ be an irreducible polynomial of degree $d$. I have to show that $f(t)$ divides $t^{p^d}-t$. The hint in this practice exam is to consider the quotient ring $F[t]/f(t)$. Can someone give me an idea of how to start even thinking of this, or how I can imagine the quotient ring?

Answer 1

Not knowing your background, I'll refer only to basic properties of fields and field extensions. First, it follows immediately from the definitions that for any field $F$, if $f\in F[t]$ is irreducible of degree $d$, the quotient $E=F[t]/(f)$ is a field extension of degree $d$. If $F=\mathbf Z/p$, then $E$ is a $\mathbf Z/p$-vector space of dimension $d$, hence is finite of cardinal $p^d$. Next, conclude by looking at the multiplicative group $E^*$ (hint: use Lagrange's theorem) .

Answer 2

Hint: $F[t]/f(t)$ is a field of characteristic $p$ and of dimension $d$, as a vector space over $\mathbb{Z}/p$, so its elements verify $t^{p^d}-t=0$. Look your favorite reference on finite fields. Deduce that the image of $t^{p^d}-t$ in $\mathbb{F}[t]/p(t)$ is zero.