How do I prove that a polynomial F[x] of degree n has at most n roots

It's a really basic question,in these days, I've been thinking why a polynomial $p(x)\in F[x]$ ($F$ a field) with degree $n$ can have at most n roots. It seems easy to prove, but I've been trying to prove this since yesterday, maybe I forgot some important details necessary to prove, I don't know. I'm solving some questions about Field Theory and I noticed that almost every question I should use this theorem, I need help.

Thanks

It follows immediately from the following lemma

Lemma $P(a)=0 \Leftrightarrow x-a | P(x)$.

Now if $P(X)$ has at least $n+1$ roots, it follows that $P(X)$ is divisible by $(x-a_1)....(x-a_{n+1})$..

Use the factor theorem and induction. This obviously only works if you're working in an integral domain.